Special equipment: a 2 or 2 1/4-inch round cookie cutter
For the dough: Whisk together the egg with 1 cup warm water in a measuring cup or small bowl. Whisk together 3 1/4 cups of the flour with 1 teaspoon salt in a large bowl. Using a wooden spoon, stir the egg mixture into the flour mixture until all of the flour is moistened but shaggy. Gather the dough and transfer to a floured work surface. Knead the dough, adding between 1/4 cup and 1/2 cup additional flour, until it is smooth, moist and soft, about 5 minutes. Wrap with plastic wrap and let rest at room temperature for 45 minutes.
For the filling: Put the chicken, garlic, onion and 1 teaspoon salt in a bowl. Stir in 3 tablespoons cold water to make a smooth paste.
To make the pelmeni: Place a small bowl of cool water by your work surface. Divide the dough into 4 pieces. Wrap 3 pieces in plastic wrap and set aside. Using a heavy rolling pin, roll the dough out 1/16-inch thick on a floured work surface. Cut out a few rounds using a 2-inch round cookie cutter. (Keep the rest of the dough covered with plastic wrap while you fill these.)
Working with 1 round at a time, brush a little water on one edge of the round. Put 1/2 teaspoon of filling in the center of the round and fold the dough over. Press out any air, pushing the filling up toward the fold, and press to seal. The dough should stretch a bit at the corners so the pelmeni is slightly crescent-shaped. Now pick up the pelmeni and pull the 2 ends together in a ring, so that the border curls up a bit. Pinch the ends together. The dumpling should resemble a tortellini. Place it on a baking sheet and repeat with the remaining ingredients; transfer batches to the freezer, uncovered, as you work.
When the pelmeni are frozen fairly solid, at least 30 minutes, transfer to a resealable bag or airtight container. Freeze for at least 3 hours before cooking.
To cook the pelmeni: Bring a large pot of salted water to a boil over high heat. Reduce the heat to a low boil and add 24 to 32 frozen pelmeni. Stir for about 15 seconds, then simmer until the meat is cooked and the dough is tender, 5 to 8 minutes.
While the pelmeni cook, put a pat of butter into each of 4 bowls. Use a slotted spoon to transfer the pelmeni directly to the bowls-don't shake the water off. Drizzle a spoonful of the cooking liquid on top, and serve with vinegar, sour cream and dill, if desired.
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