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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.
Roast drumsticks for approximately 30 to 40 minutes in an oven thatâs preheated to 425 F. To make a one-dish meal, roast chicken drumsticks on a bed of sliced or cubed potatoes and flavorful vegetables such as chopped kale, sliced onions and minced garlic.op the potatoes and vegetable with drumsticks seasoned with salt and pepper. Cover the roasting pan with foil and roast it for approximately 20 minutes at 450 F before uncovering the pan. Roast the mixture for another 30 minutes until the vegetables are tender and the chicken is done.