Step 1: Remove the giblets from the body cavities of the chickens and set aside for another use. Remove and discard the fat just inside the body and neck cavities. Rinse the chickens, inside and out, under cold running water and then drain and blot dry, inside and out, with paper towels. Cut the chickens in half: Remove the backbone with poultry shears or a sharp knife; fold the chicken open like a book and turn it over; remove the breastbone and white cartilage, then cut off the wingtips.
Step 2: Arrange the chicken halves on a baking sheet (or baking sheets) large enough to hold them and sprinkle on both sides with the rub. Pat the rub onto the chickens with your fingertips. Let the chickens sit in the refrigerator, covered, for as little as 15 minutes or as long as 4 hours; the longer they stand, the richer the flavor will be.
Step 3: Set up the grill for indirect grilling and preheat to medium. When ready to cook, brush and oil the grill grate. If using charcoal, toss the wood chips on the coals. If using a gas grill, place the wood chips in the smoker box and preheat the grill to high; then, when smoke appears, lower the heat to medium.
Step 4: Arrange the chicken halves, skin side up, on the hot grate over the drip pan. Indirect grill for 40 to 50 minutes, mopping the chickens often.
Step 5: After 30 minutes, skin should be cooked enough to start mopping.
Step 6: Set aside 1 cup of barbecue sauce for serving. When the chickens are nearly done cooking, brush the skin side with barbecue sauce and turn the halves and grill for 5 minutes to sizzle the sauce, then brush the bone side with sauce and turn the chickens again.
Step 7: Grill another 5 minutes before removing the chickens. The chickens are done when an instant-read meat thermometer inserted into the thickest part of a thigh (but not touching the bone) registers about 170 degrees.
Step 8: Transfer the chicken halves to plates or a platter and let rest for 3 minutes before serving with the reserved sauce.