In a large mixing bowl, combine the flour, egg, warm water, vegetable oil, and a pinch of salt, kneading until a smooth, soft dough forms.

Cover the dough with a clean kitchen towel and let it rest at room temperature for 30 minutes to relax the gluten.

While the dough rests, heat 1 tablespoon of butter in a skillet over medium heat and sauté the onions and mushrooms until browned, about 8-10 minutes.

Stir in the drained sauerkraut, season with black pepper, and cook for another 5 minutes to meld the flavors, then set aside to cool completely.

On a floured surface, roll out the dough to about 1/8-inch thickness and use a 3-inch round cutter or glass to cut out circles.

Place a spoonful of the cooled filling in the center of each dough circle, fold it in half, and pinch the edges tightly to seal the dumpling.

Bring a large pot of salted water to a boil and cook the pierogi in batches for 3-4 minutes, or until they float to the surface.

Remove the boiled pierogi with a slotted spoon and let them drain briefly on a wire rack or plate.

Melt the remaining 3 tablespoons of butter in a large frying pan over medium-high heat until it begins to bubble.

Fry the boiled pierogi for 2-3 minutes on each side until they turn golden brown and crispy on the outside.

Serve the hot pierogi immediately with a side of sour cream and a sprinkling of fresh herbs if desired.