Prepare chicken for even cooking: Lay chicken on a flat surface, breast-side up. Using kitchen shears or a sharp knife, cut through the breastbone, then all the way down to the opening at the other end. Open the chicken like a book, exposing the cavity, and flatten it; the backbone should crack a bit, which is normal.
Brine: In a 1-gallon container, combine water, salt, and sugar. Seal and shake to dissolve salt and sugar. (Alternatively, stir with a whisk.) Place brine into 2-gallon container, followed by the chicken; make sure the chicken is fully submerged. Cover with plastic wrap and refrigerate 15–24 hours, optimally 20 hours.
Smoke the chicken: After chicken has been brining for 20 hours, light the smoker and heat to 250 degrees F. Remove chicken from brine, and place skin side up on a flat surface. Pat dry with paper towels. (Chicken will be about ½ pound heavier from absorbing brine.) Flip over and dry the other side.
Sprinkle BBQ Spice Rub on both sides of the chicken, then rub it into the chicken, making sure the entire bird is covered. Place chicken onto the grill, cavity side down. Cover and cook undisturbed for 2 hours. Add charcoal and use vents as needed to maintain a temperature of 250 degrees F through the duration of cooking.
After 2 hours, flip chicken over so it's skin side down. Cover and cook for 20–30 more minutes. (Removing the lid of the smoker will reduce the temperature, so bring it back to 250 degrees F by adding more charcoal.)
After 20–30 minutes, check chicken for doneness by gently tugging on the legs. They should be loose, but not pulling away from the bird. A digital thermometer placed into the thickest part of the thigh, not touching the bone, should register 165 degrees F.
Carving the chicken: Cut chicken in half along the backbone. Cut each half in half again, so the breast and wing (white meat) form one serving and the drumstick and thigh (dark meat) form another. Serve.