Place 1 1/4 lb chicken breast, raw on a small rimmed baking sheet with paper towels under and over it. With a mallet or closed fist, pound the chicken to even thickness. Add a wire rack under the chicken on the baking sheet, then season both sides with 1 1/4 tsp kosher salt. Set aside in the refrigerator, uncovered, to dry brine for at least 6 hours or up to 24 hours.
Mix 2 tbsp brown sugar, 1/2 tsp each: smoked paprika, ground mustard, cumin, garlic powder, and 1/4 tsp each: chili powder, black pepper on a plate. Press the dry-brined chicken breasts into the dry rub mixture to coat on both sides. Return the chicken to the wire rack and baking sheet. Refrigerate, uncovered, for an additional 1-3 hours.
After the chicken has rested in the dry rub, pat both sides dry with a paper towel again and rub with 1/2 tbsp olive oil. Set aside at room temperature while heating the grill.
Pre-heat the grill to 450-500°F. Rub the grill grates with vegetable oil or spray with non-stick grill spray. Add the prepared chicken, "nice" side down, and cook for 6 minutes.
Flip the chicken, then brush with 1/3 cup Sweet Baby Ray's Original Barbecue Sauce. Lower the heat to about 400°F and continue to grill for 5-7 minutes or until the chicken is cooked through. Set aside to rest for 5 minutes before serving.