COMBINE sugar, ketchup, rum, lime juice, butter, ginger, coffee granules and allspice in small saucepan. Bring to a gentle boil; stir until coffee and sugar are dissolved. Remove from heat; set aside.
TRIM chicken of any visible fat. With sharp knife, make two diagonal slits about 1/2-inch deep on the top side of each breast.
COMBINE garlic, oil, salt and pepper. Rub mixture all over chicken. Arrange slit side up on rack in shallow baking pan. Brush chicken with glaze; reserve remaining glaze. Refrigerate for 30 minutes.
BROIL chicken for 5 minutes. Turn chicken over; coat with reserved glaze. Broil for an additional 4 to 5 minutes. Turn chicken over again; coat with glaze. Broil for an additional 2 minutes or until tops are sizzling and brown.