Soak the dried mushrooms in hot water for 20 minutes, then drain and finely chop them.

Sauté half the chopped onion in 1 tablespoon of butter until soft, then add the sauerkraut and mushrooms; cook for 10 minutes and season with pepper. Let cool completely.

In a large bowl, mix flour and salt, then make a well and add the egg, sour cream, and warm water.

Knead the dough on a floured surface for about 5 minutes until smooth and elastic, then cover and let rest for 15 minutes.

Roll the dough out thinly (about 1/8 inch) and cut into circles using a round cutter or glass.

Place a spoonful of filling in the center of each circle, fold over to create a half-moon, and pinch the edges tightly to seal.

Bring a large pot of salted water to a boil and cook pierogi in batches; they are done when they float to the top (about 3-4 minutes).

In a large skillet, melt the remaining butter and sauté the rest of the onion until golden brown.

Add the boiled pierogi to the skillet and fry on both sides until crispy and golden.

Serve hot topped with the caramelized onions.