In a large bowl, combine the flour and 1/2 teaspoon of salt. Make a well in the center and add the egg and warm water. Mix until a dough forms.

Knead the dough on a floured surface for about 5 minutes until smooth and elastic. Cover with a towel and let it rest for 30 minutes.

While the dough rests, prepare the filling. Melt 1 tablespoon of butter in a skillet over medium heat. Add the mushrooms and half of the diced onions. Sauté until the onions are soft and mushroom liquid has evaporated.

Add the drained sauerkraut to the skillet and cook for another 5 minutes to meld the flavors. Season with black pepper and remaining salt. Remove from heat and let cool completely.

Roll out the rested dough on a floured surface to about 1/8 inch thickness. Use a round cutter or glass to cut out circles (approx 3 inches in diameter).

Place a small spoonful of the filling in the center of each circle. Fold the dough over to create a half-moon shape and pinch the edges tightly to seal. Use a fork for a decorative edge if desired.

Bring a large pot of salted water to a boil. Drop the pierogi in batches. Cook until they float to the top, then boil for an additional 2 minutes. Remove with a slotted spoon.

In a large frying pan, melt the remaining butter over medium-high heat. Add the remaining onions and the boiled pierogi.

Fry the pierogi for 3-4 minutes per side until golden brown and crispy. Serve hot with a side of sour cream.