Allow the chicken to sit on the counter and come to room temperature for about 30 minutes while you prepare the grill. Preheat a gas or charcoal grill over indirect medium-high heat (about 375–400°F). To do so with a charcoal grill, place the charcoal in the briquet holders on either side of the grill, leaving the center open. If you don’t have briquet holders, you can just bank the coals to one side to create a hot zone and a cooler indirect zone. Place foil drip pans in the middle of the grill below the grates (these help to catch the drippings from the meat as it cooks). With a gas grill, turn one side of the burners to medium-high, leave the other side off. Whether you’re using a charcoal grill or a gas grill, this two-zone set up is essential for cooking the chicken over indirect heat so that you don’t burn the chicken before it’s cooked through.
Add a chunk of apple wood (or other fruit wood) to the charcoal briquets, or use applewood smoke pellets with a pellet smoker tube in a gas grill, if desired. This is optional, but it gives the chicken a nice, subtle smoked flavor.
Pat the chicken dry with paper towels. Rub on all sides with olive oil and sprinkle with seasoning of choice.
Place the chicken on the grill grates (over the foil drip pan, if using).
Close the lid of the grill and cook over indirect heat until the internal temperature of the chicken breast reaches 160-165°F, about 1 hour, 15 minutes to 1 hour, 30 minutes.
Baste with the barbecue sauce during the final 10 minutes, if desired.
Allow the chicken to rest for at least 10-15 minutes before carving and serving. Serve with additional bbq sauce, if desired.