To prepare the chicken: Rinse and pat dry the chicken pieces. Sprinkle on the rub generously.
To prepare the fire: Using a chimney starter, light 40 to 50 pieces of good-quality lump charcoal. When the coals are glowing, transfer them from the chimney to one side of the grill. (If you don't have a chimney starter, stack the charcoal around some crumpled newspaper in a pyramid in the grill and light the newspaper. The coals will be hot in 20 to 30 minutes)
If you have some pieces of apple or oak hardwood, feel free to add a couple to the stack of coals. Put a small foil or metal pan full of water next to the coals. Position the grilling grate so that one of the holes is over the coals so you can add coals and wood chips as needed.
When the coals are about 90% white, position the pieces of chicken, skin side up, on the grill anywhere except directly over the coals. Cover the grill with the lid, making sure that the air vent is opposite the fire. Cook the chicken for about 30 minutes, maintaining a temperature of 230° to 250°F by adjusting the vents. (Opening the vents lets in more oxygen and raises the temperature.) Add more charcoal if the temperature drops below 230°F. You'll likely need to add 15 to 20 pieces about 30 minutes after putting the chicken on.
If you're using a gas grill: Get one side of the grill hot and arrange the chicken on the other side. Close the lid and maintain the temperature of the grill between 230° and 250°F.
If your grill—gas or charcoal—didn't come with a thermometer, you can set an oven thermometer on the grate near where the chicken is cooking.
After a half hour or so, baste the chicken with some of the apple juice. Continue to cook the chicken until it's cooked through—this will take about 3 hours—basting it and checking the temperature of the grill every 45 minutes or so. As the chicken cooks, you can move the pieces around the grill if those closest to the fire seem in danger of overcooking. But keep the chicken skin side up for the duration.
Check for doneness with an instant-read thermometer after 2-1/2 hours. Cooked chicken should read 165°F in the meatiest part of the thigh or breast. You'll also know the chicken is done when its juices run clear after being sliced into with a knife.
When the chicken is cooked, pour some of the barbecue sauce into a separate container (to avoid contaminating the whole batch) and brush it onto the chicken. Cook it an additional few minutes so that the sauce adheres to the chicken in a sticky glaze; watch the chicken carefully at this point and pull it off the grill if the sauce starts to burn.
Remove the chicken from the grill and serve with some of the barbecue sauce on the side, if you like.