Prep the chicken. Place the chicken on a large baking sheet and let it come to room temperature for 20 to 30 minutes. Pat each piece completely dry with paper towels.
Make the dry rub. In a small bowl, combine the kosher salt, garlic powder, smoked paprika, chili powder, onion powder, and black pepper. Stir to combine. Sprinkle the dry rub over all the chicken pieces and use your hands to coat every surface - including under the skin.
Prep the grill. Set up your grill for two-zone cooking. Turn on the burners on one side and leave the other side off (for a gas grill). For a charcoal grill, pile coals on one side and leave the other empty. You want one hot side and one cool, indirect-heat side.
Grill the chicken. Place all chicken pieces on the indirect heat side of the grill, close to the hot side. Cover and cook for 30 to 35 minutes, flipping every 10 minutes, until an instant-read thermometer reads 160°F to 165°F (71°C to 74°C) in the thickest part of the meat.
Add the BBQ sauce. Brush the chicken generously with BBQ sauce. Cover and cook for 3 to 4 minutes. Flip the chicken, brush the other side with BBQ sauce, and cook for another 3 to 4 minutes. Repeat this process until the chicken reaches an internal temperature of 175°F (79°C).
Serve. Transfer the chicken to a serving platter. Serve immediately with extra BBQ sauce on the side.