Place 1/2 cup milk in a cup or small bowl. Stir in sugar and sprinkle yeast over the top. Set aside until foamy, about 10 minutes. Pour the remaining milk into a large bowl.
Add the melted butter, egg, salt and 1 cup of flour to the large bowl with the milk. Stir in the yeast mixture. Mix in flour 1 cup at a time until dough pulls away from the sides of the bowl and doesn't stick to your hands. Cover the bowl loosely and set in a warm place to rise for about 1 hour. The dough should almost triple in size.
For the bun style:
Place the risen dough onto a floured surface and gently form into a long snake about 2 inches wide. Cut into 1-inch pieces and roll each piece into a ball. Flatten the balls by hand until they are 4 to 5 inches across. Place a spoonful of the filling in the center and fold in half to enclose. Pinch the edges together to seal in the filling.
For the thin dough style:
Roll out the pastry as thinly as possible (less than one-eighth inch thick). Using a 3 1/2-to-5-inch cookie cutter, cut the dough into rounds. We used a 4-inch cookie cutter to produce 30 rounds. The dough will shrink after cutting. You may roll out the circles or rounds to make them larger or you may stretch them carefully by hand. Beat the egg with the water. Brush the top of each pastry round with the egg mixture. These can be either baked for 20 to 25 minutes or fried until golden brown.In an large, heavy skillet or deep fryer, heat the oil to 375 degrees F (190 degrees C). Deep fry the piroshki in batches until golden brown on one side; gently turn and fry the other side. Remove and let drain on a plate lined with paper towels.
Use about 2-3 tablespoons of filling for each circle of dough. Shape the filling into an oval and place it on half of the circle of dough. Fold the other half of the circle of dough over to enclose the filling. Press the edges of the dough with the fingers or the tines of a fork to seal. Brush the tops with egg mixture to seal.
Arrange the filled pieces on a lightly greased baking sheet.
Place in the oven and bake 20- 25 minutes , or until golden brown
Soft and fluffy dinner rolls stuffed with a savory filling
Place filled piroshki, seam-side down on a greased baking sheet
.** Repeat with the remaining dough, spacing the rolls 2 inches apart. Brush the tops with the beaten egg. Set them aside to rise until puffy and nearly doubled (30 minutes). When the rolls are nearly finished proofing, preheat the oven to 375F