Roll out dough and press into three 9-inch round baking pans. (Reserve some of the dough to form lattice top on pies.) Preheat oven to 350 degrees.
Peel, slice, and cook squash until tender. Saute onions in oil for 10-15 minutes. Mix onions with mushrooms and add squash. Continue to saute until all vegetables are soft. Add the remaining ingredients, except paprika, and mix.
Pour mixture over dough in baking pans. Sprinkle some paprika for color. Use reserved dough to form lattice design on top of pies. Brush pies with beaten egg.
Bake pies for 30-40 minutes.
Note: Pies can be decorated around rim of baking pan with a thin braid formed of dough.