In a large bowl, mix the lukewarm milk, sugar, and yeast together. Let the mixture sit at room temperature for about 10 minutes.
To the milk mixture, add water, eggs, oil, salt, and flour. The dough will be runny and sticky and that's exactly what you want. Knead it by hand (wet your hands to prevent sticking) or with a stand mixer until the dough turns elastic.
Place the dough in a bowl covered with plastic wrap or in an airtight container and let it rest in a warm place overnight.
With oiled hands, break the dough into small pieces. Flatten out the dough and place the filling inside. Next, flatten out the filling inside the dough.
Fold over the sides and pinch the edges together to seal the filling inside the dough. Once it's sealed tight, shape the dough into a long oval shape.
Fill a deep frying pan about halfway with oil over medium heat.
Fry each side of the piroshky until golden brown on each side and lay them out on a paper towel to extract the excess oil.
Cabbage Filling
Dice the onions and carrots and shred the cabbage. Preheat a large skillet with oil and cook the onions and carrots until they soften. Next, add in the shredded cabbage and cook for an additional 5 minutes.
Season to taste with salt and pepper and stir the butter in until it melts.
Meat Filling
Sauté the ground chicken with salt and pepper over medium-high heat until lightly brown and fully cooked. Add 1/2 of the diced onion and cook it until softened.
Stir the other half of the raw onion into the meat mixture. Send it through the meat grinder. Stir to combine and set aside.