Heat oil in a large nonstick skillet over medium heat. Add onion to pan. Cook 5 minutes or until lightly browned; stir occasionally. Transfer onion to a bowl.
Combine cheese and next 4 ingredients (through egg white). Divide turkey mixture into 4 equal portions, shaping each into a 1/2-inch-thick patty. Return pan to medium heat. Coat pan with cooking spray. Add patties to pan; cook 4 minutes or until brown. Turn patties over; cook 1 minute. Top each patty with 1 cheese slice; cook 3 minutes or until cheese melts and patties are done.
Place bread slices in a single layer on a baking sheet; broil 2 minutes or until toasted. Spread 1 tablespoon mustard on each bread slice; top each serving with 1 patty. Divide onion mixture evenly among sandwiches.