Heat oil in a large skillet. Add sliced onions and cook for 30 minutes, stirring frequently, or until golden brown. Add raspberry preserves, brown sugar and mustard to skillet; stir until combined. Bring to a simmer and cook 3 minutes or until slightly thickened. Set aside 1 cup of mixture.
Preheat grill. Grill chicken, bone side up, for 15 minutes. Turn and baste with sauce. Grill an additional 15 minutes, basting with sauce, or until chicken is done. Serve with reserved cup of sauce.