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Heat oven to 400 degrees F.
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Lightly oil a large baking dish. Season the drumsticks with salt and pepper and arrange in the dish in a single layer. Drizzle over a little olive oil and bake uncovered in the hot oven for 20 minutes.
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Prepare the glaze in the meantime. Mix the last 6 ingredients together in a small bowl until evenly combined.
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Take the chicken out of the oven after 20 minutes and pour over just enough of the glaze to coat each drumstick with a little run-over in the pan. Return to the oven and bake for another 30 to 40 minutes, turning several times, until the chicken is tender and nicely glazed.
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Let the chicken rest for a few minutes before serving.
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You may save any leftover glaze in the refrigerator as long as it didn't come in contact with the raw chicken.