Heat a gas or charcoal grill to medium-high (300-350°F) heat.
Place the coals to one side or leave one side on low. Combine the rub ingredients in a spice grinder or mortar and pestle. Grind into a fine powder.
Coat the chicken pieces with olive oil. Toss with the rub until evenly coated. Place the chicken on the side of the grill set with the lower heat, cover and grill about 40 to 50 minutes, flipping once, or until the chicken is almost cooked. Brush with sauce, flip and grill on the hottest part of the grill for 3 to 5 minutes. Brush the other side and repeat.
Let rest 5 to 10 minutes before serving. Accompany with additional sauce.
Here’s How: Alabama White Sauce
Combine 1 cup mayonnaise, 2 tbsp sugar, 1/ 4 cup apple cide vinegar, 1 tbsp whole grain mustard, 1 tsp horseradish, 1/ 4 tsp cayenne and a few dashes of hot sauce in a bowl.
Whisk until well blended. Seasons with salt, pepper and hot sauce to taste.