Step 1Cut stem and root end off zucchini and slice into 1/8-inch-thick rounds. Cut eggplant into 3 slices lengthwise, then cut into 1/2-inch slices lengthwise. Holding slices together, cut slices into 1/2 inch pieces crosswise. Set chunks aside. Cut onion into 1/4 inch slices lengthwise. Place onion flat side down on cutting board and cut into 1/8-inch pieces. Wipe off mushrooms and break off stems. Cut stems in half or thirds and set aside. Cut mushroom caps into 1/4-inch strips. Holding strips together, cut crosswise into half-inch pieces.
Step 2Place oil in large skillet and tilt to coat bottom of pan. Heat over medium-high heat and stir in vegetables, spreading over skillet bottom. Season with salt and pepper to taste. Stir until lightly browned, about 6 to 8 minutes.
Step 3Stir eggs in bowl with fork until they are all one color. Add milk and stir briskly to blend well.
Step 4Grease bottom and sides of a 13x9-inch baking dish. Place single layer of bread over bottom. Top with half of mixed vegetables. Sprinkle with one-third of cheese. Repeat by covering with another layer of bread and spreading vegetables and sprinkling with another third of cheese. Cover with the third layer of bread. Drizzle egg mixture over top layer of bread. Sprinkle with last one-third of cheese. Cover with plastic wrap and refrigerate at least 3 hours or overnight.
Step 5Bake at 350 degrees until top is lightly browned and toothpick inserted in center is moist with crumbs, but not wet, 30 to 35 minutes.