Position knife blade in food processor bowl; add first 9 ingredients. Pulse 12 times, scraping sides of processor bowl once. Place 1/3 cup onion mixture in a heavy-duty, zip-top plastic bag; reserve remaining onion mixture. Add chicken to bag; seal bag, and marinate chicken in refrigerator 1 hour, turning bag occasionally.
Remove chicken from marinade, discarding marinade. Coat grill rack with cooking spray; place rack on grill over medium-hot coals (350° to 400°). Place chicken and thyme sprigs, if desired, on rack; grill, covered, 4 minutes on each side or until chicken is done. Discard thyme sprigs. Serve chicken with reserved onion mixture.