Coat a large skillet with cooking spray, and place over medium heat until hot. Add onion; cover and cook 10 minutes or until deep golden, stirring frequently.
Combine salt, thyme, red pepper, and black pepper, and stir well. Sprinkle half of spice mixture over onions. Add 1/4 cup of chicken broth to the onion mixture, and cook for 10 minutes, stirring frequently. Add 2 tablespoons broth, and cook until liquid evaporates, scraping bottom of skillet with a wooden spoon to loosen the browned bits.
Repeat procedure with remaining 2 tablespoons broth. Add vinegar, and cook an additional 2 minutes or until most of the liquid evaporates. Remove onion mixture from pan; set aside, and keep warm.
Rub chicken with remaining spice mixture. Recoat skillet with cooking spray, and place over medium-high heat until hot. Add the chicken, and cook for 5 minutes on each side or until done.
Serve the caramelized onion mixture over chicken.