Combine butter with 1/3 of the water in a saucepan and bring to a boil. Add remaining water.
Sift flour into a bowl, add butter-water mixture, and mix. Knead mixture into a smooth dough for 5 minutes, put on a plate, and cover with an inverted bowl. Rest for 30 minutes.
Place quartered cabbage in a pot, cover with water, and boil until cabbage is soft, about 15 minutes. Drain cabbage and wrap in a clean kitchen towel. Squeeze out as much liquid as possible.
Cut dry cabbage into thin slices and put in a bowl. Add onion and season with caraway, salt, and pepper. Cook in a non-stick pan until softened, about 10 minutes. Add fresh mushrooms and onions and cook for another 10 minutes. Mix cream and potato starch in a cup; add to mushrooms. Bring to a boil and season with salt and pepper.
Roll about 1/4 of the dough out on a floured surface about 1/16-inch thick, and cut rounds about 3 1/2-inches across using a cookie cutter or a glass. Add leftover dough to the next 1/4 dough and repeat process.
Place about 1 teaspoon of cabbage filling in the center of each dough round, fold over into a half-moon shape, and press and seal the edges with a fork. Set the filled pierogies aside under a towel to prevent drying.
Bring a large saucepan of water to a boil, and carefully drop about 15 pierogies at a time into the boiling water. Boil in simmering water until they float to the top, about 5 to 10 minutes. Cook remaining pierogies the same way in batches.