Combine the flour and salt in a large bowl, forming a well in the middle.
Add the butter and milk and mix lightly until the flour is absorbed. (Add a little warm water as needed. ) Knead until the dough sticks together, cover and allow to rest for a few minutes, then knead until smooth.
Cover and set aside. (This dough should be somewhat soft, since more flour will be added as the dough is rolled out. ) Or, in a processor, combine the flour, salt, and butter, stirring a few times, then with the machine running, add liquids until a ball forms.
Allow to rest for a few minutes and then process until smooth.
Place the dough in a lightly oiled bowl, turn, and cover for about 30 minutes. (It may be wrapped in plastic and refrigerated for a day or so, brought to room temperature and then rolled out. ) Place a fourth of the dough on a floured work surface, re-covering the rest of the dough.
Roll into a circle, starting at the center and rolling outwards to maintain an even thickness, turn over and again from the center, roll the dough out to about 1/8-inch thickness.
Run a hand under the bottom to loosen it.
With a 3-inch biscuit cutter or wine glass, cut into rounds.
When done cutting, place 1 Tb of the filling on one side of each, flipping the other half over the top and sealing with your fingers or a fork.
Make sure each is sealed or the filling will come out when cooked.
Place each of the dumplings on a floured cookie sheet, keeping the dumplings covered with a towel.
Repeat with the remaining dough, saving the scraps until last.
Use as little flour as possible in this process or the scraps will be come tough and heavy.
Gently drop 12 to 15 varenyky into 3 quarts of boiling water, in a large wide pot or Dutch oven.
Do NOT overcrowd.
Stir and cook for about 3 to 4 minutes.
Drain in a colander and place on a lightly oiled cookies sheet, shaking to coat with a thin film to prevent sticking.
Do NOT pile the dumplings on top of each other as this distorts their shape.
If the varenyky are to be frozen, remove with a slotted spoon when they float to the top.
DO NOT overcook.
Place in the freezer on an oiled cookie sheet when tepid (lukewarm).
When they are rigid, store in tightly sealed plastic bags.
TO SERVE: Varenyky may be poached, pan-fried or steamed, served as an accompaniment for meats or served with sour cream, chopped sauteed onions or fried bacon with a little bacon fat.