Mash brown sugar, garlic, salt, and pepper together in a mortar and pestle to form a paste. Spoon paste into a resealable plastic bag. Add chicken, coat with paste, squeeze out excess air, and seal the bag. For best results, marinate in the refrigerator at least 1 hour to overnight.

When ready to cook, heat oil in a small saucepan over medium heat. Add onion and sauté until softened, about 5 minutes. Stir in ketchup, vinegar, and Worcestershire sauce. Bring to a simmer and cook until flavors blend, about 10 minutes.

Meanwhile, preheat an outdoor grill for medium heat and lightly oil the grate. Remove chicken from the bag.

Cook chicken on the preheated grill until brown, 3 to 5 minutes, turning once halfway through. Turn off one of the burners and move chicken over indirect heat. Maintain the temperature at 300 to 350 degrees F (150 to 175 degrees C).

Cook chicken for 20 minutes, turning halfway through. Baste with sauce, then continue to cook, basting every 10 minutes, until chicken is no longer pink at the bone and the juices run clear, 20 to 25 more minutes. An instant-read thermometer inserted near the bone should read at least 165 degrees F (74 degrees C).