Pour a cup of warm milk in a bowl, add a pinch of salt and the egg, then whisk together.
In a separate bowl, mix together flour and dry yeast.
Start adding the flour with the yeast mixture into the bowl with wet ingredients. To make the job easier, use an electric mixer with a hook attachment. Add ½ cup of flour at a time and mix until the dough forms into a ball. You might need less than 3 cups of flour, that’s the reason why we don’t add it all at once.
When the dough is formed, cover the bowl with a plastic wrap or a towel and place in a warm place for the dough to rise. Leave it for 2 hours or until the dough doubles in size and is soft and easy to work with.
While the dough rises, prepare the potato filling.
Peel the potatoes and onion. Cut potatoes in pieces, place them in a pot, cover with cold water, and cook covered for 20 minutes until potatoes are tender.
Meanwhile, dice the onion and sauté in butter until onion is soft.
Mash potatoes with a potato masher, add salt, pepper, sautéed onions, and ½ cup of water and mix well. The filling is ready to use.
Now you can start shaping your piroshki. Pinch off pieces approximately the size of a golf ball. (Keep the rest of the dough in a bowl covered while working on each piece)
Roll the piece into a disk shape, about 3 ½ inches in diameter, adding flour when needed.
Place a tablespoon of the filling on the center of the disk, fold disk over the filling and firmly pinch the edges to seal.
Place formed piroshki on a lightly floured surface and let them sit for about 5-10 minutes.
In a large heavy skillet heat the oil and fry piroshki in batches on both sides until golden brown (sealed side down). Remove and drain piroshki on a plate lined with a paper towel.
You can serve piroshki hot or cold, they taste great either way!