Add the first 6 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. Punch down. Cover and let rest for 10 minutes.
Meanwhile, in a shallow pan combine mushrooms, zucchini, peppers, and garlic. Toss vegetables with 1 tablespoon olive oil. Bake in 450 degree F oven 20 minutes or until tender and brown on edges, stirring occasionally. Transfer to a large bowl. Stir in pasta sauce; set aside.
Grease the bottom and side of a 9-inch springform pan. Shape one-fourth of the dough into a small ball. Shape the remaining dough into a large ball. On a lightly floured surface, roll the large ball into a 16-inch circle. Place the dough into the bottom and up the side of the prepared pan, allowing dough to extend slightly over edge. Sprinkle with 3/4 cup of the cheese. Top with vegetable mixture. Sprinkle with the remaining 3/4 cup cheese.
Roll the small ball of dough into a 9-inch circle; place on top of vegetable mixture. Brush dough with some of the milk. Fold edge of bottom dough over top dough. Press lightly to seal. Brush with remaining milk. Cut slits in dough for steam to escape.
Bake in a 350 degree F oven for 50 to 60 minutes or until golden and edge sounds hollow when tapped. Cool in pan on a wire rack 20 minutes. To serve, use a metal spatula to loosen crust from pan; remove side of pan. Cut into wedges.