Place the chicken pieces and 1/2 cup BBQ sauce in a large Ziploc bag. Seal the bag, pushing out the air. Shake or massage the bag until the chicken is coated. Let the chicken rest in the BBQ sauce in the fridge for one hour.
Near the end of the marinade time, preheat the oven to 425 degrees F.
Wash the sweet potatoes and cut off and discard a small piece from each end. Cut into small cubes. Set aside.
Chop the broccoli into florets. Set aside.
Prepare the spice mix by adding the salt, garlic powder, chili powder, and cumin in a small bowl. Mix to combine.
Place the sweet potato cubes into a large bowl and sprinkle with one tablespoon of olive oil. Toss to combine. Using your hands is the most efficient way to do this.
Sprinkle with half of the spice mix. Add about one quarter of the mixture, stir, and add another quarter to be sure that the potatoes are evenly covered.
Lay the chicken pieces skin side down on a rimmed baking sheet.
Surround the chicken with the prepared sweet potatoes. Spread the potatoes out in a single layer.
Roast the chicken for 25 minutes.
When 25 minutes have elapsed, use tongs to flip over the chicken so it is skin side up.
Remove the chicken and add 1/2 tbsp of BBQ sauce to the tops of each piece of chicken. You can use a spoon or a pastry brush.
Roast for an additional 20 minutes.
Add the broccoli florets to the bowl and sprinkle with the other tablespoon of olive oil. Toss to combine.
Sprinkle the rest of the spice mix over the broccoli, adding one half, stirring, and adding the other half.
Add the broccoli to the chicken and potatoes, placing them in a single layer.
Continue cooking for 15 minutes.
Test the chicken with an instant-read thermometer. It is cooked when the internal temperature of the chicken has reached 165 degrees F.