In a large bowl, whisk eggs, milk, salt and pepper. Stir in cubed sourdough bread, bacon, mushrooms, 1 1/2 cups mozzarella, scallions and fresh thyme. Transfer to a buttered 3-qt. baking dish; top with 1 cup mozzarella. Cover; refrigerate overnight. Bake at 350 degrees until just set, about 1 hour.