Cut sourdough rolls into 1-inch cubes; layer evenly in a 13- x 9- x 2-inch baking dish coated with cooking spray. Set aside.
Coat a large nonstick skillet with cooking spray; place over medium-high heat until hot. Add sausage, and cook until sausage is browned, stirring until it crumbles; drain if necessary. Layer sausage and cheese over bread cubes.
Add mushrooms and onion to skillet; cook over medium-high heat, stirring constantly, 4 minutes or until tender. Layer over cheese in baking dish; set aside.
Combine milk and remaining 4 ingredients, stirring until smooth. Pour over bread-mushroom mixture in baking dish. Cover and chill at least 8 hours or overnight.
Bake, uncovered, at 350° for 35 minutes or until a knife inserted in center comes out clean. Let stand 5 minutes before serving.