Place flour in a large bowl; make a well in the center. Place egg, sour cream, cooking oil, and salt in well. With your fingertips, combine ingredients, gradually adding the 1/3 cup water. On a lightly floured surface, knead the mixture until smooth and elastic. Divide dough into fourths; cover with a towel.
On a lightly floured surface, roll one portion of dough 1/16 inch thick. Cut into 4-inch circles. Spoon a rounded teaspoon of Cheese Filling or Mushroom Filling off center on each circle. Moisten edges of dough. Fold dough over the filling to form a half-circle. Seal edges by pressing with tines of a fork. Transfer to a lightly floured baking sheet and cover loosely with a towel. Repeat with remaining portions of dough.
In a Dutch oven, bring 12 cups of lightly salted water to boiling. Add about 1/3 of the pierogis. Cook for 4 minutes. Remove with a slotted spoon to a rack or to paper towels to drain. Keep warm while cooking the remaining pierogis. Then, serve drizzled with melted butter. (Or, heat 1 tablespoon butter in an extra-large skillet and cook drained pierogis about 1-1/2 minutes per side or until lightly browned, adding more butter as needed.) Makes about 36 pierogis.
In a skillet, cook chopped onion in butter until tender. Add chopped mushrooms, salt, and pepper. Cover and cook for 10 minutes, stirring occasionally. Remove from heat; cool slightly. Stir in bread crumbs and beaten egg yolks. Cool filling before using. Makes about 1-1/2 cups.
In a food processor bowl or blender container, cover and process or blend dry cottage cheese or ricotta cheese until smooth. (If using a blender, add 1 egg yolk before processing.) Add 1 egg, 1 egg yolk (omit egg yolk here, if using blender), salt, and white pepper. Cover and process or blend until just combined. Makes 1-1/2 cups.