Prepare the Brine: Combine water, salt, brown sugar, and crushed garlic in a large container. Stir until dissolved. Submerge the chicken in the brine, cover, and refrigerate for 2–4 hours.
Make the Spice Rub: Mix brown sugar, paprika, cumin, onion powder, garlic powder, smoked paprika, salt, and black pepper in a small bowl.
Remove and Dry the Chicken: Take the chicken out of the brine, discard the brine, and pat the chicken dry with paper towels.
Season the Chicken: Rub the spice mix generously over all sides of the chicken pieces.
Preheat the Grill: Heat the grill to medium heat (350–375°F). Oil the grates to prevent sticking.
Grill the Chicken: Place chicken on the grill and cook over indirect heat for 20–25 minutes, flipping occasionally. Brush BBQ sauce onto the chicken during the last 5–10 minutes of cooking. Move to direct heat briefly to caramelize the sauce.
Check for Doneness: Use a meat thermometer to check internal temperature:
Breasts: 165°F. Thighs/Drumsticks: 175°F. Rest the chicken for 5 minutes before serving.