In a small bowl, dissolve the sugar in the warm milk and water (about 110째F or 45째C). Sprinkle the yeast over the top and let it sit for about 5-7 minutes, or until it becomes frothy.
In a large bowl, combine the flour and salt. Make a well in the center.
Add the yeast mixture, eggs, and vegetable oil into the well.
Mix until a dough forms, then knead on a floured surface for about 5-7 minutes, or until smooth and elastic.
Place the dough in a lightly oiled bowl, cover with a damp cloth or plastic wrap, and let it rise in a warm place for about 1-2 hours, or until doubled in size.
In a skillet, heat the vegetable oil over medium heat. Add the chopped onion and cook until translucent, about 5 minutes.
In a bowl, combine the cooked ground meat, mashed potatoes, cooked onions, salt, and black pepper. Mix until well combined.
Punch down the risen dough and divide it into 16-20 equal pieces.
Roll each piece into a small circle about 4 inches in diameter. Place a spoonful of filling in the center.
Fold the dough over the filling to form a half-moon shape, pinching the edges together to seal. You can also form small round pockets by folding the dough in half.
Preheat your oven to 375째F (190째C). Line a baking sheet with parchment paper.
Place the pirozhki on the prepared baking sheet. Brush the tops with the beaten egg mixed with water.
Bake in the preheated oven for 20-25 minutes, or until golden brown.
Enjoy your meal!