From Time/Life's Foods of the World - Russian Cooking, 1969:

The most typically Ukrainian flour-based dish...is probably the dumplings called varenyky. The dish is permanently listed on all restaurant menus, there are cafés that serve nothing else, and in most households it is made once or twice a week. Everyone in the Ukraine knows that good varenyky must be made with a dough that is tender and thin, yet has enough strength and elasticity to encase the filling without bursting when it boils. the number of "right" ways to achieve this ideal, however, must very nearly equal the number of good cooks in [the Ukraine]. Unanswered questions and open disputes abound. some cooks use nothing but wheat flour for their varenyky dough, while others add buckwheat flour, mashed potatoes or cottage cheese beaten or sieved to creamy smoothness. The liquid for the dough may be milk or water or a mixture of both (which raises another question - cool or lukewarm?), with or without either whole eggs or egg yolks. Does the addition of a pinch of sugar, baking powder, cream of tartar or a spoonful of melted fat improve the dough? To complicate the matter, country women who learned to make varenyky two or three generations ago make yeast-dough dumplings light as thistledown. Once made, the dough is rolled thin (but should it be rolled on a cloth or a board?), then cut into squares or circles three or four inches wide. A spoonful of the filling is placed slightly off-center on each piece, the dough is folded over to make a triangle or half-moon, and the edges, which must be free of the slightest morsel of filling, are pinched together - very firmly. Varenyky that burst are not varenyky, and the one who made them is not a cook - no question about that, at least. The well-sealed dumplings are slipped into a pot of boiling, salted water. When they float to the top they are done, and may be removed with a slotted spoon and set on a plate, preferably without overlapping. If the dough is right, the filling will taste right. It may be sauerkraut, mashed potatoes, cabbage, chopped mushrooms, ground meat, fish or - most common of all - dry cottage cheese mixed with egg. To complete the dish, the varenyky are garnished with buttered crumbs, browned chopped onions or bits of crumbled bacon, and eaten with liberal dollops of sour cream.

It was at this point that I realized that varenyky are simply the Ukrainian version of pierogies from Poland, which were covered very well by Todd, thanks to his wife's wonderful recipe:

http://foodsoftheworld.activeboards.net/forum_posts.asp?TID=1626&title=perogies-wifes-recipe-with-a-brief-pictorial - http://foodsoftheworld.activeboards.net/forum_posts.asp?TID=1626&title=perogies-wifes-recipe-with-a-brief-pictorial

FoTW went on to provide some info about another, splendid personality of these little pockets of joy:

Visiting friends at a dacha, or summer cottage, outside Kiev we had dessert varenyky with a fruit filling. Our hostess picked plums from a dooryard tree. While they boiled in honey and water, she casually tossed together a varenyky dough of four cups of flour, two of water and three egg yolks. In less than 15 minutes, 10 people were enjoying meltingly tender dumplings, each containing a single plum like a precious jewel in rich purple syrup. The mystique of varenyky dough, I said to myself, was surely all nonsense. Just to be sure I asked my friend, "You just use flour and water and egg yolks. Nothing more?" "Absolutely nothing. But of course the flour must be ground from hard wheat, and of course freshly drawn spring water is the best liquid." The eggs, I gathered, could be either brown- or white-shelled.

Culinaria Russia provides some additional information:

Almost every Ukrainian adores the...dough pockets, filled with whatever the fields, forests and gardens have to offer....The savory versions have pâté de foie gras, potato, cabbage, bean, onion, pork crackling or mushroom filling. The fans of sweet varenyky like them with poppy seeds, apples, plums or prunes. And who can resist varenyky with fresh cherries or Quark?....They can be enjoyed as an appetizer, main dish, an entrée or as a dessert. Some people like to eat varenyky the next day fried brown and crispy in butter.

Other fillings listed by Culinaria include ground beef, pork or lamb, yogurt and raisins, vanilla-flavoured cream and lemon peel, forest or garden berries, jam or poppy seeds. The fruit for varenyky filling are boiled with sugar or honey and a little water or juice to make a compote. Both Culinaria and FoTW caution not to let the varenyky stick together, either while boiling or when removed from the pot. To help with this, toss them in a bit of melted butter straight out of the pot, and keep the separated when laid out on a platter.

Below is Time/Life's recipe for cheese-filled Varenyky, served as a dessert. For a "non-dessert" cheese varenyky, simply omit the sugar. If there is any interest, I can also provide recipes for filling using fruit (cherries or other), sauerkraut, cabbage, and a combination of liver and bacon. For mashed potato filling, follow the instructions found in Todd's outstanding tutorial (link above).

To make about 16 dumplings:

Dough

2 cups all-purpose flour
1 egg
1/2 cup milk
1 teaspoon salt

Cheese Filling:

1 pound large-curd cottage or pot cheese
2 tablespoons sugar
1 egg yolk
1 tablespoon melted butter, cooled
1/2 teaspoon salt
2 cups sour cream (divided)
4 tablespoons melted butter, hot (reserved)

Dough: Pour the flour into a large mixing bowl and make a deep well in the center. Drop in the egg, milk and salt. With your fingertips or a large spoon, slowly mix the flour into the liquid ingredients, then mix vigorously until the dough is stiff enough to be gathered into a compact ball. If the dough crumbles add an additional 1 to 2 tablespoons of milk to make the particles adhere. Dust the ball with flour, wrap in wax paper, and chill for 30 minutes.

Cheese Filling: Purée the cheese in a food mill, or rub it with the back of a spoon through a fine sieve set over a bowl. Then beat in the sugar, egg yolk, melted butter (1 tablespoon) and salt, and stir in 1 cup of the sour cream, 1/4 cup at a time. Continue to stir until the ingredients are thoroughly mixed. Taste for seasoning; if you prefer the filling sweeter, stir in additional sugar.

On a lightly-floured surface, roll the dough out into a circle about 1/8-inch thick. cut out as many circles as possible with a 3.5- to 4-inch cookie cutter. Then gather the remaining scraps into a ball, roll out again and, and cut out additional circles. With a pastry brush, coat each circle with a light film of the beaten egg white. Drop 1 tablespoon of the cheese on the lower half of each circle.  Bring the exposed half of the circle up over the filling and press the edges all around firmly with the back of a fork. Make certain that the edges are thoroughly sealed to prevent the filling from seeping through. Set aside, loosely covered with a towel, until ready to cook.

Bring 4 quarts of salted water to a boil in a 5- to 6-quart pot and drop in 6 dumplings. Lower the heat and simmer the dumplings uncovered for 8 to 10 minutes, or until they float to the surface of the water. Remove with a slotted spoon and transfer to a heated platter. Cover loosely with foil to keep them hot while you cook the remaining dumplings similarly. Moisten with melted butter and serve with the remaining sour cream.

Here is an "alternate recipe" for varenyky dough, as well as a few suggestions for different fillings, from Culinaria Russia:

For the Basic Dough:

3.5 cups (500 grams) all-purpose flour
1 cup (250 millilitres) water
2 eggs
1 teaspoon salt
Butter and sour cream

Make a dough from the flour, water and eggs and leave to rest for 20 minutes. Roll out to about 1/16-inch (2 millimetres) thick (double the thickness for varenyky with cherries). Use a thin-walled glass to press circles out of the dough. Place a teaspoonful of filling in the centre of each circle, fold them in half and press the edges together well. Cook the filled dough pockets in a generous quantity of lightly salted, boiling water for about 10 minutes. When they float to the surface, add a glass of cold water, bring back up to a boil and remove with a slotted spoon. Drizzle with melted butter straightaway so that they do not stick together and serve with sour cream.

Savory Cabbage Filling:

The filling for savory cabbage varenyky contains...chopped white cabbage, onion, salt and pepper as well as a little lemon juice if preferred. They are served with melted butter or also sour cream.

Generous 1 pound (1.5 cups) (500 grams) cabbage, finely chopped
Oil for cooking
1 large onions, finely chopped
1 medium-sized carrot, finely chopped
1 medium-sized parsley root, finely chopped
Salt and pepper

Sauté the cabbage for 20 minutes. Sauté the other vegetables together and then add to the cabbage, season with salt and pepper. Cook everything together for about 15 minutes. These varenyky are usually served with chopped onions, browned in a lot of butter.

Liver and Bacon Filling:

Generous 1 pound (500 grams) ox liver, prepared ready for cooking
4 tablespoons (1/4 cup) butter or oil
3 medium-sized onions, finely chopped
4 thick slices (100 grams) fatty bacon
Salt and pepper

Roughly chop the liver, fry in 2 tablespoons fat and cook until soft. Sauté the onions separately in 2 tablespoons fat. Put the liver and bacon through a grinder, combine with the onions, season with salt and pepper.

Cherry Filling:

Varenyky with cherries are a welcome summer feast - whether as a sweet main dish or a warm, fruity dessert. They are filled with cherries and sugar and served with melted butter or also with cherry sauce. Cherry varenyky are the top favourites among fruity varieties, but the sweet dough pockets also taste good with blueberries. Other forest and garden berries, plums, jam or poppy seeds also make for tasty fillings.

1.75 pounds (750 grams) cherries, pitted
2/3 cup (150 grams) sugar

For the sauce:

1 handful of cherry pits
2/3 cup (150 grams) sugar

Combine the cherries with the sugar and leave to stand in a warm place for a couple of hours. Then drain and keep the juice. Fill the varenyky with the sugared cherries. For the sauce, crush the cherry pits in a mortar and pestle, bring to a boil in 3/4 cup (200 millilitres) water, simmer for about 10 minutes and then strain. Add the sugar to the liquid, bring to a boil again and reduce over high temperature. Combine the sugar syrup with the cherry juice and serve with the varenyky.

A few more suggestions from Culinaria Russia:

Varenyky with a meat filling can be prepared with ground beef, pork or lamb. They are served with onions [carmelised in butter]. Leftovers from the Sunday roast are also a good option for stuffing the delicious little dough pockets.

Sweet Varenyky with Quark, or yogurt, and raisins are a treat for varenyky fans. They are filled with vanilla-flavoured cream and lemon peel or, if preferred, are flavoured with a little rum and are also served with sour cream.

The wealth of mushrooms in the Ukraine of course means that a wide variety of mushroom varenyky are served. For the filling the mushrooms are sautéed with onions and seasoned with herbs according to preference. Sour cream is the best accompaniment for this variety of varenyky.