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1. To make the dumplings, lightly beat the yogurt, egg, and salt together in a mixer. Slowly add the flour, beating until smooth. Knead the dough on a floured surface until smooth. Add water only if needed. Form into a ball, wrap in a kitchen towel and set aside in a cool place for 2 hours.
2. For the filling, peel and quarter the potato. Cover with cold water, bring to a boil, lower heat and simmer 15-20 minutes until it offers no resistance when pricked with a fork. Drain and set aside. Slowly melt 2 tablespoons of the butter. Whip the potato with the butter until it is fluffy and creamy. Set aside.
3. Heat the remaining butter in a pan. Add the onions and saute until they are translucent. add the mushrooms, salt and pepper and continue to saute until the mushrooms are tender and the moisture has evaporated, about 5 minutes. Remove from pan and let cool slightly.
4. Mix the mashed potatoes with the mushroom mixture.
5. Bring a large pot of salted water to a boil. Roll out the dough until it is 1/8 inch thick, then cut into 3-inch rounds with a cookie cutter or the top of a glass. Put 1 heaping teaspoon of filling on one side of the circle of dough. Fold the other side over the filling so the pirogi looks like a half moon. Press the edges together and seal with thumb and index finger.
6. Lower heat until water gently simmers. Carefully place pirogi in the water. 1 at a time, in batches of 4 to 6, for 5-7 minutes. Remove with a slotted spoon and drain on paper towels in a warm place. Serve with fried onions, crispy shallots, or sour cream.