Set up your grill with two zones—a hot zone for direct heat and a cool zone for indirect heat—then pat the chicken drumsticks dry with paper towels.
Coat the drumsticks evenly with the avocado oil.
In a small bowl, mix the salt, pepper, smoked paprika, garlic powder, onion powder, and dried thyme.
Rub the seasoning mix thoroughly over all the drumsticks.
Place the seasoned drumsticks on the cool zone.
Close the lid and cook for 15–20 minutes.
Turn the drumsticks and continue cooking, with the lid closed, for an additional 10–15 minutes. (This indirect heat ensures the chicken cooks completely without burning.)
Move the drumsticks to the hot zone for 3–5 minutes to get a nice char and crisp the skin.
Lightly brush the drumsticks with BBQ sauce.
Grill for 1–2 minutes to allow the sauce to set into a glaze.
Serve immediately.