Lightly brush grill rack with vegetable oil. Then prepare grill with low-heat coals, or heat gas grill.*
Sauté onion and garlic in oil in skillet until softened, 3 minutes. Mix onion, garlic, vinegar, ketchup, mustard, Worcestershire, molasses and hot-pepper sauce in small bowl. Season chicken with salt and pepper.
Grill, uncovered, 5 minutes, until lightly browned. Turn over; spoon sauce on chicken. Grill 5 minutes. Turn over; spoon on more sauce. Cover grill. Grill 10 minutes. Spoon on sauce. Turn over every 5 to 10 minutes, until internal temperature registers 175 degrees F on instant-read thermometer, about 50 minutes more. Do not spoon on sauce during last 10 minutes of cooking. Makes 4 servings.