These Polish dumplings are traditionally served with soft, browned onions or sour cream.
Remove stems from shiitake mushrooms, and trim stems of white and cremini mushrooms. Place mushrooms in a food processor, and pulse about eight times, until finely chopped. Set aside.
In a medium saute pan, melt butter and oil over low heat; add shallots. Cook, covered, until softened, about 2 minutes. Add mushrooms, lemon juice, salt, and pepper, and mix well. Raise heat to medium, and cook, stirring occasionally, until mixture is nearly dry, about 8 to 10 minutes. Add cream, thyme, and parsley, and cook about 1 minute. Transfer to a medium bowl.
To form dumplings, place a heaping teaspoon of the filling toward the front of a circle of pierogi dough. Moisten the edges of the dough with water, and fold in half around filling to form a semicircle, firmly pinching the edges closed. Set aside on a lightly floured baking sheet. Continue to fill and form pierogi, keeping them covered with plastic wrap, until all the dough has been used.
Meanwhile, bring a large pot of salted water to a boil, and add half the pierogi. Boil gently until they float to the surface, about 2 minutes, and continue cooking for another 3 to 4 minutes. Drain, and repeat with remaining pierogi. Serve immediately.