Butter bottom and sides of a 9"x13" baking dish.
Butter bread and cut into cubes. Set aside.
Beat milk, eggs, salt, and black pepper together in a large bowl. Mix in bread cubes, 1 1/4 cup cheddar cheese, ham and vegetables.
Pour egg mixture into baking dish; cover and refrigerate 8 hours or overnight.
Preheat oven to 350 degrees.
Sprinkle remaining cheese (1/2 cup) on top of dish. Bake uncovered egg mixture in the preheated oven until center is set, about 1 hour. Allow to sit and cool for 15 minutes before serving.
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