Butterfly the chicken. With a large fork, poke several holes in the underside of each chicken half--this helps the brine to soak deeply into the meat. Combine the Coke and salt in a large resealable plastic bag, stirring to dissolve the salt. Place the chicken halves in the brine, seal the bag, and refrigerate for 2 hours. Preheat the grill to high.
Combine the rub ingredients in a bowl. Remove the chicken from the brine and pat dry with paper towels (discard the brine). Rub the paste over the chicken, then let it stand for 15 minutes.
Turn 1 side of the grill to low (maintain high heat on the other side). Arrange both chicken halves on the low-heat section, skin side down. Top the halves with a cast iron skillet and a brick. Cover and grill for 10 minutes, then rotate the chicken 45°. Reposition the skillet and brick on the chicken; grill for 10 more minutes, or until skin is crispy and brown.
Flip the chicken over and grill, without weight, for 20-30 minutes, or until it's 180° in the thigh.
Serve chicken with dipping sauces.