Melt the butter in a saucepan over medium heat, then add the minced garlic and cook for 1 minute until fragrant.
Add the ketchup, brown sugar, chili sauce, Worcestershire sauce, yellow mustard, celery seed, salt, black pepper, and cayenne to the pan; bring to a boil, then reduce heat and simmer 2–3 minutes to combine. Remove from heat and set aside.
Heat the grill to medium and lightly oil the grates to prevent sticking.
Place the cut-up chicken on the grill and cook for 30–40 minutes, turning occasionally so it cooks evenly.
Baste the chicken with the prepared sauce, then continue grilling, basting every 5 minutes, for about 15 more minutes until the chicken is cooked through and glazed.
Remove the chicken from the grill and serve with any leftover sauce.