Whisk eggs, milk, salt, and black pepper together in a bowl.
Spray a 2-quart baking dish with cooking spray. Spread bread pieces out in an even layer on the bottom of the prepared dish; top with sausage, Monterey Jack cheese, egg mixture, and Cheddar cheese, respectively. Cover the baking dish with plastic wrap and refrigerate, 8 hours to overnight.
Preheat oven to 325 degrees F (165 degrees C). Remove casserole from refrigerator and let sit at room temperature for 15 minutes.
Bake casserole in the preheated oven until bubbling and hot throughout, about 45 minutes.
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