Make the dough: Stir the flour and salt in a mixing bowl. Add the egg and half of the water. Use a wooden spoon or your hands, and mix well to combine. Add the remaining water and mix until a dough comes together. Turn the dough onto a lightly floured surface and knead until the dough is smooth and homogenous, about 2 minutes. Shape the dough into a ball, cover with plastic wrap, and set aside at room temperature for at least 2 hours. 

Make the filling: Add the beef, onion, salt, and pepper to a large mixing bowl and mix until the ingredients are well-incorporated. Set aside.

Prepare the dough: Place the dough on a heavily floured surface. Knead the dough until it becomes pliable and less sticky, about 2-4 minutes. Use a rolling pin to roll out the dough into a thin circle about 15-inches in diameter. Sprinkle the dough with flour and fold the top edge over the bottom to create a semi circle, and then fold the semi circle into a quarter of a circle. Let it rest under plastic wrap for 5 minutes.

Unfold the dough back into a circle and sprinkle it lightly with flour. Using a rolling pin, roll the dough out as thinly as possible until the circle has roughly doubled in size. Sprinkle the dough with flour again. Next, fold the dough into an accordion shape about 1½ inches deep by folding it back and forth horizontally in layers approximately 2 inches thick. Using a sharp knife, cut the folded dough crosswise into 2-inch pieces. Unfold each piece into long rectangular strips, then stack these strips horizontally on top of each other. Finally, cut the stacked dough crosswise into 2-inch squares. Cover the dough squares with a towel or plastic wrap to prevent drying.

Fill and shape the dough: Place 1 teaspoon of the beef filling into the center of a piece of dough. Take the top edge of the square farthest from you and fold it over towards you, creating a triangle, and just cover the filling, leaving a ¼ inch border of dough on the bottom of the piece of dough. Use your fingers to pinch and seal the dough over the filling. Pick up the pelmeni, holding each bottom corner of the dough with your thumbs and pointer fingers. With one of your middle fingers, press in the center of the meat-filled dumpling and fold one edge of dough over the other with your thumbs. Seal these two edges together tightly to create the pelmeni’s signature shape, a ring shaped dumpling with a pouch on the bottom. Repeat filling and shaping the rest of the pelmeni.

Freeze pelmeni on a plate if making for later and after 20 minutes, transfer to a freezer bag. To eat immediately, bring a large pot of salted water to a boil. Place all the pelmeni into the pot and boil for 3-4 minutes until they float to the top and are cooked through.

Serve hot and lightly drizzle with white vinegar. Pelmeni can be stored in an airtight container in the refrigerator for 3 days.