In a large skillet, saute the zucchini, red pepper, mushrooms and onion in oil until tender. Add the garlic, thyme, salt and pepper; saute 1 minute longer.
In a 13-in. x 9-in. baking dish coated with cooking spray, layer half of the bread cubes, zucchini mixture, goat cheese and Parmesan cheese. Repeat layers.
In a small bowl, whisk the eggs, milk and nutmeg. Pour over top. Cover and refrigerate overnight.
Remove from the refrigerator 30 minutes before baking. Bake, uncovered, at 350° for 45-50 minutes or until a knife inserted near the center comes out clean. Let stand for 10 minutes before cutting. Yield: 12 servings.