I tasted a recipe similar to this at a friend's house once, which is what inspired me to try this. These buns take some time to make but it is well worth it - especially since you get leftovers that freeze well!
2 packages active dry yeast
1/2 cup warm water (110-115 degrees)
3/4 cup warm milk (110-115 degrees)
1/4 cup sugar
1/4 cup shortening (OR 1/4 c. butter + additional 3/4 t. salt)
1 teaspoon salt
3 1/2 - 4 cups flour
1 pound ground beef
1/4 cup chopped onion
8 oz. tomato sauce
8 slices American cheese, quartered
In a mixing bowl, dissolve yeast in warm water. Add 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 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-inch circle with two pieces of cheese and about 3 tablespoons beef mixture. Bring dough over filling to center; pinch 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 degrees for 8-12 minutes or until golden brown. Serve warm.