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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.
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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.
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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.
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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.
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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.
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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.
