Heat oil in large ovenproof skillet over medium heat. Add bacon and cook, stirring, until browned, about 4 minutes. Add asparagus and onion; cook, stirring, until softened, about 5 minutes. Transfer to plate and keep warm.
Spray same skillet with nonstick spray. Arrange slices of bread in bottom of skillet, fitting them snugly and overlapping as needed.
Whisk together eggs, egg whites, milk, mustard, salt and pepper in medium bowl; pour over bread, allowing milk mixture to soak in. Sprinkle bread with 1 tablespoon Parmesan. Set skillet over medium heat and cook, shaking pan frequently, until egg mixture is almost cooked through, about 5 minutes.
Place skillet under broiler 5 inches from heat and broil until egg mixture is set, about 2 minutes. Remove skillet from broiler (leave broiler on); top bread with bacon-vegetable mixture, Cheddar and remaining 1 tablespoon Parmesan. Return to broiler until Cheddar is melted, about 2 minutes.