Use a small paring knife to carefully cut alongside the bone and open the meat outward. Pat the drumsticks thoroughly dry with paper towels and lightly coat them with avocado oil and Cajun seasoning across the skin and exposed meat.
Air Chill for 30 Minutes
Once seasoned, place the drumsticks uncovered in the refrigerator for 30 minutes while you fire up the grill.
Start Over Direct Heat
Start cooking the seasoned drumsticks over direct heat to develop color and char on the exposed meat and skin, turning and repositioning drumsticks as needed.
Move to Indirect Heat
Once the drumsticks develop the color you want, move them to the indirect side and maintain the grill at 275-300°F. Continue cooking until the thickest part of the drumsticks reaches 185-195°F, chicken beside the bone without touching it.
Make Cajun Butter Sauce
While the drumsticks are still on the grill, combine unsalted butter, honey, Cajun seasoning and minced garlic in a grill-safe pan and place it directly on the grill, whisking gently until the sauce is smooth.
Glaze, Finish & Rest
Brush Cajun honey butter on the drumsticks during the final few minutes of grilling. Transfer to a platter, brush with some more Cajun honey butter sauce and rest for 5 minutes before serving.
Video
Notes
Butterfly every drumstick and apply the glaze in several thin coats instead of one heavy layer.
Keep flipping the drumsticks on the grill after you've sauced them. Honey can burn quickly.
Allow the sauced drumsticks to rest for 5 minutes so the juices can redistribute before serving.