Heat the pellet grill or smoker to 300℉ with your choice of wood or pellets. *See notes on cooking temps.
Prepare the chicken for the smoker. Pat the skin of the chicken dry with paper towels. Then season the whole chicken with Texas-style rub first. Next add a layer of your favorite barbecue seasoning.
Place large pieces of celery and/or carrot in the bottom of a half size foil pan, lining them up flat for the chicken to rest on top. *This keeps the chicken elevated off the bottom of the pan to allow more air circulation, and it allows you to catch the drippings to be used for the sauce. Place the seasoned chicken on top of the bed of vegetables.
Smoke the chicken on the pellet grill or smoker at about 300°F until the internal temperature reaches about 155-160°F. *About 2 hours for us, but cooking times will vary. Always cook to the correct internal temp, not a certain time.
While the chicken is cooking on the grill, prepare the pan sauce. Add finely diced onion and butter to a small saucepan and heat over medium low heat until onions are tender. Then add the remaining sauce ingredients, reduce heat to low, and simmer for about 15 minutes, stirring occasionally.
Once the chicken reaches about 155-160℉ internal, add the pan sauce to the aluminum foil pan with the chicken. Then put back on the smoker and continue to cook until the internal temperature reaches about 175-180°F. *NOTE – Your chicken is technically done when the internal temp of the thickest part of the chicken breasts reaches 165°F, but for pulled chicken I like to cook to a higher temp to make the meat easier to shred.
When the chicken is done, transfer to a cutting board or large baking sheet to rest for about 20 minutes. While the chicken is resting, pour the pan juices and sauce from the foil pan into a bowl. Allow the fat to come to the top of the liquid, then skim off with a spoon and discard.
Use two forks or your hands to shred and pull the chicken, discarding any gristle or unpleasant bits as you shred. Then spoon some of the reserved pan juices over the shredded chicken, mixing everything together. *We added about ⅓ cup of the juices, but you can adjust to your preference.
Add a drizzle of barbecue sauce (optional) and serve the pulled chicken on buns for a pulled barbecue chicken sandwich, or use the shredded chicken as desired for other recipes.