Coat a 9-by-13-inch baking dish with cooking spray.
Cook sausage in a skillet over medium heat, crumbling with a fork, until browned. Transfer to a bowl.
Add oil, onion and bell pepper to skillet; cook, stirring occasionally, until vegetables soften, about 5 minutes. Add sausage and cook, stirring, until vegetables begin to brown, about 5 minutes more. Remove from heat and set aside.
Whisk eggs and egg whites in a large bowl until blended. Whisk in milk, mustard, salt and pepper. Stir in 1/3 cup Cheddar.
Arrange bread in a single layer in prepared baking dish. Pour egg mixture over bread and top with reserved vegetables and sausage. Sprinkle with remaining 1/3 cup Cheddar. Cover with plastic wrap and refrigerate for at least 5 hours or overnight.
Preheat oven to 350°F.
Bake casserole, uncovered, until set and puffed, 40 to 50 minutes. Serve hot.
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