Step 1

Combine 1 cup water and dried mushrooms in a saucepan over low heat, bring to a simmer, and cook until tender, about 10 minutes. Drain mushrooms, reserving cooking water. Chop mushrooms.

Step 2

Heat olive oil in a separate saucepan over medium heat and cook onion until soft and translucent, about 5 minutes. Add mushrooms and sauerkraut and mix well; season with salt and pepper. Pour in the cooking water, cover, and simmer until water has evaporated and sauerkraut is soft, 15 to 20 minutes.

Step 3

Place flour in a large bowl and make a well in the center. Crack egg into the well and add enough warm water, 1 tablespoon at a time, until a dough forms, stirring constantly with your hands. Knead well, continuing to add more water as needed. Knead until dough is soft and smooth.

Step 4

Dust a work surface with flour. Cut off 1/4 of the dough and roll out to a thickness of 1/8 inch. Cut out circles using a glass or a round pastry cutter.

Step 5

Fill each dough circle with 1 teaspoon of sauerkraut filling. Fold dough over into a half-moon shape and seal edges. Cover with a clean dish towel so pierogi won't dry out and repeat with remaining dough and filling.

Step 6

Bring a large pan of salted water to a gentle boil. Add the pierogi in batches and cook until they float to the surface, 5 to 7 minutes. Cook for 2 to 3 minutes once they float to the surface. Remove with a slotted spoon and drizzle with melted butter.