In a small bowl, combine the first five ingredients. Crumble turkey over mixture and mix well. Shape into 10 thin patties. Spoon red pepper and cheese onto center of five patties; top with remaining patties and press edges firmly to seal.
Grill burgers, covered, over medium heat or broil 4 in. from heat for 5-7 minutes on each side or until a thermometer reads 165° and juices run clear. Serve on buns.