Preheat your grill to 230-250 degrees. Combine all of the basting sauce ingredients and whisk to blend.
Remove the giblets from the bird, along with any fat. Rinse the chicken inside and out with cold water. Pat the chicken inside and out with paper towels to dry it.
Put the chicken on a cutting board, breast side down with the tail pointing away from you. Pull the chicken to an upright position on its neck end. With a sharp knife, cut down the right or left side of the backbone, through the thigh joint, which is just gristle, then go down the backbone, cutting through the rib bones. Be careful to keep to the backbone, not the breast bone.
Repeat the process on the other side of the back bone.
Open the chicken, breast or skin side down, with the breast end pointing toward you. In the middle of the breast you'll see the keel bone. Some people call it the breast bone, but it's the keel bone. At the top of the keel bone, in the center, there is a white piece of gristle about the size of a quarter; with your knife, cut straight down the center of the gristle.
Place your thumbs on either side of the cut, and with your fingers under the keel bone or your fingers on the skin side of the breast, push down with your thumbs and up with your fingers and the keel bone will pop up; you can then pull the keel bone and attached gristle out. Turn the chicken over and press on the breast, flattening the chicken out.
Season with pepper and salt.
Place the chicken on your pit or grill skin side up and cook indirectly for 3 to 4 hours, depending on the size of the chicken, or until the internal temperature in the thighs are 165 degrees or higher. Do not turn the chicken. Baste the open side of the chicken with the sauce at 1 1/2 hours, and baste the skin side every 30 minutes until done.