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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.
5Boil chicken drumsticks for 30 to 40 minutes. Since drumsticks have less meat on them than thighs, they do not usually need to cook for quite as long. 6Give a whole chicken about 1 hour to cook.[2] A standard 4-lb (1800-g) chicken will need a full hour to cook when boiled.