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1. In a large mixing bowl, combine the cold water, kosher salt, and brown sugar. Stir well to dissolve the salt and the sugar. Pour the mixture into a gallon-size zip-top bag and add the chicken drumsticks. Seal the bag and refrigerate for at least 11/2 hours, turning the bag occasionally.
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2. While the chicken sits in the brine, prepare the Smoked Paprika Rub. Combine all ingredients in a small bowl and set aside.
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3. Next, prepare the Spicy Apple Glaze. Combine all of the ingredients, including the vanilla bean and seeds, in a 6-quart saucepan and bring to a boil over medium-high heat, stirring often to dissolve the sugar. Once the mixture has come to a boil, reduce the heat to medium and cool until it is reduced to about 1 cup and has a glaze consistency, 30 to 35 minutes. Remove the pan from the heat, and strain the glaze through a fine-mesh sieve into a bowl. Discard the solids. Set the glaze aside until ready to use.
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4. Remove the chicken from the brine (discard the brine) and pat the drumsticks dry with paper towels. Season on both sides with the Smoky Paprika Rub and let sit at room temperature for 30 minutes.
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5. While the chicken is resting, preheat the grill to a medium-high heat.
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6. Spray the grates of the grill with nonstick cooking spray or brush them with olive oil. Place the drumsticks on the grill and sear for 2 to 3 minutes on each side. Once they are seared, move them to a cooler part of the grill and close the lid. Cook for about 20 minutes, or until cooked through. During the last 5 to 10 minutes of cooking, brush the drumsticks all over with the Spicy Apple Glaze. You can apply multiple coats of the glaze if you like.
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7. Remove the drumsticks from the grill to a clean plate and cover with foil if you're waiting a few minutes to eat. Serve with an extra drizzle of the Spicy Apple Glaze, if desired.
