Mix the flour and salt. Make a well, add the egg and warm water, and knead until a smooth dough forms, about 6 minutes. Cover and rest for 30 minutes.
Mix the pork, beef, onion, garlic, black pepper, and a pinch of salt until just combined.
Shape the Pelmeni:
Roll the dough thinly on a floured surface. Cut rounds with a 3-inch cutter.
Place about 1 teaspoon filling in each round. Fold into a half-moon, press the edges firmly, then bring the corners together.
Boil and Serve:
Bring a large pot of salted water to a gentle boil. Add the pelmeni in batches and cook until they float and the filling is cooked through, 5 to 7 minutes.
Lift with a slotted spoon and toss with melted butter. Serve with sour cream, dill, and black pepper.
Notes
Resting the dough makes it easier to roll.
Do not overfill the dumplings or the seams may open.