In a large mixing bowl, dissolve yeast in warm water. Add the milk, sugar, shortening, egg, salt and 2 cups flour; beat until smooth. Stir in enough remaining flour to form a soft dough.
Turn onto a floured surface; knead until smooth and elastic, about 4-6 minutes. Place in a greased bowl, turning once to grease top. Cover and let rise in a warm place until doubled, about 30 minutes.
In a large skillet, cook beef and onion until meat is no longer pink; drain. Stir in tomato sauce. Remove from the heat; set aside.
Punch dough down; divide into 16 pieces. On a lightly floured surface, gently roll out and stretch each piece into a 5-in. circle. Top each circle with two pieces of cheese and about 3 tablespoons beef mixture. Bring dough over filling to center; inch edges to seal.
Place seam side down on a greased baking sheet. Cover and let rise in a warm place until doubled, about 20 minutes. Bake at 400° for 8-12 minutes or until golden brown. Serve warm. Refrigerate leftovers.