Drain sauerkraut.

Rehydrate dried mushrooms in 1 cup boiling water for at least an hour. Drain, reserving liquid. Rinse and coarsely chop rehydrated mushrooms. Coarsely chop fresh mushrooms.

In a large sauce pan, melt 2 tablespoons of butter over medium-high heat. Add fresh mushrooms and cook until tender and golden, about 10 minutes.

Increase heat. Add sauerkraut, rehydrated mushrooms, and reserved liquid. Bring to a boil. Reduce heat, cover, and simmer, stirring occasionally, for 20 minutes, or until sauerkraut is tender.

If needed, increase heat and boil off liquid. Season with salt and pepper. Remove from heat and stir in remaining 2 tablespoons of butter.

Transfer to a food processor and pulse several times to chop finely.

Fill and cook pierogi as described in basic pierogi dough.