Preheat the oven to 400°F. Lightly grease a rimmed sheet pan.
In a medium bowl (or large Ziploc bag), combine chicken breasts, 1 tablespoon of olive oil, brown sugar, 1 ½ teaspoons of kosher salt, smoked paprika, garlic powder, onion powder, ground mustard, and ¼ teaspoon ground black pepper. Mix well to coat.
Arrange the chicken in the center of the prepared baking sheet. Spread the broccoli, bell pepper, and onion around the chicken. Drizzle the vegetables with the remaining 2 tablespoons of olive oil; toss to coat. Season the vegetables with kosher salt and ground black pepper to taste.
Bake uncovered for 22-26 minutes, stirring the vegetables and basting the chicken with the bbq sauce during the final 10 minutes. Remove the chicken from the oven when the juices run clear and an instant-read thermometer registers 165°F. For a more charred, crispy finish, transfer the pan to the broiler for a few minutes at the very end. Just keep a close eye on it, as the chicken and veggies can burn easily.