Sprinkle chicken pieces with salt and pepper. Grill, skin side down, uncovered, on a greased rack over medium heat for 20 minutes. , Meanwhile, in a small saucepan, make barbecue sauce by heating oil over medium heat. Add onion; saute until tender. Stir in remaining sauce ingredients and bring to a boil. Reduce heat; simmer, uncovered, for 10 minutes. , Turn chicken; brush with barbecue sauce. Grill 15-25 minutes longer, brushing frequently with sauce, until a thermometer reads 165° when inserted in the breast and 170°-175° in the thigh.