In a large bowl, mix the flour and salt, then make a well in the center to add the warm water, egg, and vegetable oil.

Knead the dough on a floured surface for about 5-7 minutes until it becomes smooth and elastic, then cover it with a damp cloth to rest for 30 minutes.

While the dough rests, sauté the diced onion and chopped mushrooms in 1 tablespoon of butter over medium heat until softened and browned, about 8 minutes.

Stir in the drained sauerkraut and black pepper, cooking for another 5 minutes to meld the flavors, then remove from heat and let cool completely.

Roll out the rested dough on a floured surface until it is very thin, about 1/8 inch thick.

Use a round cutter or glass rim to cut out circles of dough, approximately 3 inches in diameter.

Place a small spoonful of the sauerkraut filling into the center of each dough circle.

Fold the dough over the filling to create a half-moon shape and pinch the edges tightly together to seal, ensuring no filling escapes.

Bring a large pot of salted water to a rolling boil and carefully drop the pierogi in batches.

Cook the pierogi until they float to the surface, then simmer for an additional 2-3 minutes before removing them with a slotted spoon.

In a large skillet, melt the remaining 3 tablespoons of butter over medium heat and add the sliced onion.

Cook the sliced onions until they are golden and caramelized, which should take about 10-15 minutes.

Add the boiled pierogi to the skillet with the onions and fry for 2-3 minutes per side until they are slightly crispy and golden brown.

Serve the pierogi hot, topped with the buttery caramelized onions and a dollop of sour cream if desired.