To make the Cajun rub, in a small bowl, stir together the chili powder, paprike, thyme, oregano, onion flakes, garlic flakes, sugar, and cayenne. The rub can be made up to 2 weeks in advance and stored in an air-tight container at cool room temperature. To make Cajun brine, in a sauce pan over medium-high heat, combine 3 tablespoons of the Cajun rub, the salt, and the water. Bring to a boil, stirring to dissolve the salt. Remove from the heat and let cool completely, then transfer to a large, deep, nonreactive container that will hold the turkey breast. Rinse the turkey breast halves and add them to the brine, then add enough water to cover the turkey halves completely. Cover and refrigerate for at least 8 hours or up to 24 hours. Remove from refrigerator 30 minutes before grilling. Prepare charcoal or gas grill for indirect grilling over medium heat. Oil the grill rack. Remove the turkey breast halves from the brine and pat them dry. Discard the brine.
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For a gas grill- Place a shallow pan half full of water at the edge of the grill rack away from the heat elements. Add the wood ships to the grill in a smoker box or a perforated foil packet. Place the turkey breast halves on the grill rack away from the heat elements. Cover the grill and cook the turkey breast halves, adding more water to the pan as needed, until they are a rich golden brown and are cooked through, about 1 - 1 1/2 hours. NOTE-use 2 handfuls of woodchips total.
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