In a large bowl, whisk together the flour and salt. Make a well in the center and add the egg, sour cream, and 4 tablespoons of melted butter, stirring with a fork until a shaggy dough forms.

Turn the dough out onto a lightly floured surface and knead for 5 to 7 minutes until it becomes smooth and elastic. Wrap the dough tightly in plastic wrap and let it rest at room temperature for 30 minutes to relax the gluten.

While the dough rests, melt 2 tablespoons of butter in a large skillet over medium heat. Add the finely chopped onion and chopped mushrooms, sautéing for 6 to 8 minutes until the mushrooms are browned and the moisture has completely evaporated.

Stir the chopped sauerkraut and black pepper into the skillet with the mushrooms and onions, cooking for another 5 minutes to meld the flavors. Season with additional salt to taste, then transfer the filling to a bowl to cool completely.

Roll out the rested dough on a floured surface to about 1/8-inch thickness. Use a 3-inch round cutter or glass to cut out circles, gathering and re-rolling any remaining dough scraps.

Place a rounded tablespoon of the cooled sauerkraut filling in the center of each dough circle. Fold the dough over to create a half-moon shape, pressing the edges firmly together and crimping with a fork to secure the seal.

Bring a large pot of salted water to a gentle boil. Drop the pierogi in batches of 6 to 8 and cook until they float to the top, then let them boil for an additional 2 minutes before removing with a slotted spoon to drain.

Melt the remaining 2 tablespoons of butter in a large skillet over medium-high heat and cook the sliced onions until soft and golden brown, about 10 minutes. Toss the boiled pierogi into the pan and fry for 2 to 3 minutes per side until crispy and golden brown, then serve immediately topped with the caramelized onions and a side of fresh sour cream.