Pass farmer's cheese and cooked potatoes separately through a food grinder or food processor.

Heat oil in a large skillet over medium heat and cook mushrooms and onions until soft, about 15 minutes. Remove from heat.

Combine farmer's cheese, potatoes, mushroom mixture, sour cream, salt, and pepper in a bowl. Mix together and set filling aside.

Place flour on a clean work surface and make a well in the center. Crack both eggs into the well. Add butter and a few tablespoons of the warm water. Mix with your hands, gradually adding more warm water, 1 tablespoon at a time, as you go. Knead well, continuing to add more water as needed. Knead until dough is soft and smooth, adding more flour to the work surface as needed.

Cut off 1/4 of the dough and roll out to a thickness of 1/8 inch. Cut out circles using a glass or a round pastry cutter, saving the excess dough for your next batch.

Fill each dough circle with 1 teaspoon of filling. Fold dough over into a half-moon shape and pinch edges together to seal. Cover with a clean dish towel so pierogi won't dry out and repeat with remaining dough and filling.

Bring a large pot of salted water to a gentle boil. Add the pierogi in batches, about 20 at a time, and cook until they float to the surface, 10 to 15 minutes. Remove with a slotted spoon and drain in a strainer. Repeat with the remaining pierogi.