Combine first 3 ingredients in a shallow dish; place egg whites in a bowl. Place panko in a shallow dish. Dredge 1 breast half in flour mixture. Dip in egg whites; dredge in panko. Repeat with remaining chicken.
Heat 1 tablespoon oil in a large ovenproof skillet over medium-high heat. Add chicken to pan; cook 2 minutes. Add remaining 1 tablespoon oil. Turn chicken over; cook 2 minutes. Coat chicken with cooking spray; place pan in oven. Bake at 450° for 5 minutes.
Turn chicken over; top each breast half with 2 tablespoons sauce, 2 tablespoons Parmigiano-Reggiano, and 3 tablespoons mozzarella.