In a large mixing bowl, combine the egg, sugar, and salt. Gently warm the milk and add it along with the vegetable oil. Gradually stir in flour and yeast until a wet dough forms. Knead for about 8-10 minutes.

Place dough in a lightly oiled bowl, cover with a kitchen towel, and let it rise in a warm area for 60-90 minutes.

While the dough rises, boil salted water, add whole potatoes, and cook for 15-20 minutes until tender. Drain and mash with butter, seasoning with salt and pepper.

In a skillet, melt remaining butter and sauté chopped onion for 4-5 minutes. Add ground meat and season. Cook until browned, about 8-10 minutes. Let cool.

Mix mashed potatoes with the meat mixture and adjust seasoning with dill, salt, and pepper.

Punch down the risen dough and divide it into 12 pieces. Shape into ovals, fill with mixture and seal edges. Flatten slightly.

Heat oil in a deep pot to 350°F - 375°F. Fry piroshki in batches for about 5 minutes on each side until golden brown. Drain on paper towels.