Preheat oven to 350 F. Lightly grease a 10 x 13 inch baking dish. Line the bottom and sides with 1/2-inch slices of bread. In a frying pan over medium heat, melt butter. Add the potatoes and cook 2-3 minutes. Add onion, bell pepper, and mushrooms, and continue to cook until onion is translucent and potatoes are tender, 3- 5 minutes. Layer the cooked vegetables over bread. In the same frying pan, cook the breakfast sausage until cooked through. Spread cooked sausage over vegetables in baking dish. In a medium mixing bowl, beat eggs and milk together until smooth. Stir in grated cheese. Pour egg and cheese mixture over bread, vegetable, and sausage. Allow to set 15 minutes to overnight, for bread to absorb some of the egg mixture. Cook in 350 F oven 30 to 35 minutes until egg is set. Remove from oven and allow to rest 5 minutes before serving.