1. Heat oil in a large nonstick skillet over medium-high heat. Add onion; cook 18 minutes or until golden brown, stirring frequently. Add apple and vinegar; cook an additional 5 minutes. Stir in 1/4 teaspoon salt and 1/4 teaspoon pepper. Remove onion mixture from pan; set aside.
2. Sprinkle chicken breast halves evenly with remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper. Coat pan with cooking spray; return pan to medium-high heat. Add chicken, and cook 6 to 7 minutes per side or until done.
3. Top each chicken breast evenly with onion mixture and cheese. Cook, covered, 5 minutes or until cheese melts.