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.