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Combine the balsamic vinegar, honey, brown sugar, soy sauce, rosemary sprigs, salt and garlic cloves, stirring to mix well.
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Pour into a large reseable zip-top plastic bag and add the drumsticks.
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Seal with as little air as possible.
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Place in the refrigerator and marinade for at least 2 hours.
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Preheat oven to 450 degrees.
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Place the chicken on a foil-lined baking sheet.
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Bake until the skin is carmelized and very dark in spots, about 30 to 35 minutes.
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While chicken is cooking, pour marinade in a small saucepan.
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Bring marinade to a boil then reduce the heat and simmer over low heat, about 20 minutes.
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Remove from heat.
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During the last 10 minutes baste chicken with the marinade.
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Place chicken on a serving platter and baste with the remaining glaze.
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Sprinkle with the sesame seeds and chopped parsley.