In a large bowl, combine the flour, baking powder and salt. Cut in shortening until mixture resembles coarse crumbs. Stir in milk to form a soft dough. Divide into 12 portions.
Roll each into a 6-in. circle; press onto the bottoms and up the sides of greased muffin cups. Fill each with 2 tablespoons sausage and about 1 teaspoon cheese. Sprinkle with onion. Spoon about 2 teaspoons beaten egg over each.
Pinch crust together over filling to seal; brush with butter. Bake at 400° for 18-22 minutes before removing from pan to a wire rack. Serve warm. Refrigerate leftovers.