Add the first 7 ingredients to a 1-1/2- or 2-pound bread machine according to the manufacturer's directions. Select the dough cycle. When cycle is complete, remove dough from machine. Punch down. Cover and let rest for 10 minutes.
Meanwhile, for filling, in a large skillet cook the ground beef, mushrooms, and green pepper until meat is brown. Remove from heat; drain off fat. Stir in the cheese and the pizza sauce.
Divide the dough in half. On a lightly floured surface, roll each half into a 10-inch square. Cut each square into four 5-inch squares. Divide the filling among the squares; moisten edges. Lift a corner of each square and stretch to opposite corner over filling. Seal edges by pressing with tines of a fork.
Transfer calzones to greased baking sheets. Prick tops with a fork. Brush with the milk. Do not let rise. Bake in a 425 degree F oven for 12 to 15 minutes or until golden brown. Remove from baking sheets; cool slightly on wire racks. If desired, serve calzones with additional warmed pizza sauce.