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Prepare the smoker for barbecuing, bringing the temperature to 200 to 220 degrees F. Remove the turkey breast from the refrigerator and let it sit at room temperature for about 30 minutes. Cut a 3-foot piece of cheesecloth and dampen it thoroughly with water. Wrap the breast in the cheesecloth and tie the ends.
Transfer the breast to the smoker, skin side up, and cook for 1 1/4 to 1 1/2 hours per pound, or until the internal temperature reaches 180 degrees F. Wet the cheesecloth down with more water at 30-minute intervals, or as appropriate for the style of smoker.
After 4 hours, remove the cheesecloth, snipping it with scissors if necessary, and discard. Baste the turkey with the Hot Times Mop- if possible in the smoker - every 30 minutes for the remainder of the cooking time. When the turkey is finished cooking, remove it from the smoker and allow it to sit under tented foil for at least 10 minutes before carving.