Off the heat, coat the same skillet with cooking spray. Arrange the bread tightly in the bottom of the skillet. In a large bowl, whisk together the eggs, egg whites, milk, mustard, salt, and pepper. Pour the mixture over the bread and allow it to soak in entirely (like when making French toast). Sprinkle the bread with 1 tablespoon of Parmesan cheese.
Place the skillet over medium to medium-high heat. Cook, shaking the pan frequently to promote even cooking, until almost cooked through to the top, about 4 or 5 minutes.
Place the skillet under broiler to set the eggs, about 2 minutes. Remove the skillet from the broiler and top with bacon-vegetable mixture, remaining Parmesan cheese, and Cheddar cheese. Place the skillet under the broiler again until the cheese melts, about 1-2 minutes. Slice into 6 pieces and serve.