Combine rub ingredients. Arrange chicken on rimmed baking sheet and sprinkle both sides evenly with rub. Cover with plastic wrap and refrigerate at least 6 hours and up to 24 hours.
Using a gas grill, place the disposable aluminum pans each filled with 1 1/2 cups water directly on 1 burner of gas grill (opposite primary burner). Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Turn primary burner to medium-high and turn off other burner(s). Adjust primary burner as needed to maintain grill temperature of 325 - 350 degrees.
Clean and oil cooking grate. Place chicken, skin side down. Sear 2 minutes then turn over and sear 2 minutes on second side. Move chicken to cooler part and brush both sides with 1/3 cup sauce. Arrange chicken, skin side up, with leg quarters closest to fire and breasts farthest away. Turn burner 1 off and turn burner 2 to adjust temperature to 300°F. Cover and cook for 25 minutes.
Brush both sides of chicken with remaining 1/3 cup sauce and continue to cook, covered, until breasts register 160 degrees and leg quarters register 175 degrees, 25 - 35 minutes longer. At end, to crisp skin, over medium heat cook 2 minutes on each side.
Transfer chicken to serving platter, tent loosely with aluminum foil, and let rest for 10 minutes. Serve.