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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.
Preheat a gas or charcoal grill to 400 degrees F. If you are using a charcoal grill without a temperature gauge, this will take about 20 minutes. Brush the grates of the grill lightly with vegetable oil to prevent sticking.he Best Way to Cook Drumsticks on the Grill. Brush chicken drumsticks with barbecue sauce before grilling them for extra flavor. Photo Credit amenic181/iStock/Getty Images. Chicken drumsticks are flavorful, fairly fatty cuts that stand up well to grilling, and remain moist when cooked over high heat.