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Put the chicken drumstick on the hottest part of the grill and cook them for 25 minutes, rotating them frequently with a pair of tongs to ensure even cooking. Step 5. Remove the chicken drumsticks from the grill and put them on a warm serving plate.
1 Rub drumsticks with seasonings. 2  Sear drumsticks on the grill for approximately 3 minutes per quarter turn. 3  After searing all sides, move drumsticks to a part of the grill where they can cook by indirect heat. ( 4 See notes below).