Preheat the oven to 350 degrees F.
Season the potatoes with 1 1/2 teaspoons salt and 1 teaspoon pepper and coat with 1/2-ounce of the oil. Roast the potatoes for 15 minutes.
Preheat the oven to 375 degrees F.
Mix 1 1/2 teaspoons salt, 1 teaspoon pepper, the garlic powder and paprika. Sprinkle the seasoning on the chicken, then sear skin-side-down in a hot pan with 1/2-ounce oil.
Brush the chicken with the BBQ sauce then roast skin-side-up until the internal temperature of the breast is 160 degrees F, 15 to 20 minutes. Rest the chicken for 5 minutes.
Bring a pot of salted water to a boil. Blanch the broccoli until bright green and tender, 1 to 2 minutes, then shock in an ice bath to stop the cooking and preserve the vibrant color.
Saute the roasted potatoes with the broccoli in 1/2-ounce oil, adding up to 1 1/2-ounces more oil if needed, and season with some salt and pepper.
Pour some BBQ sauce on the plate, then place the chicken on top and garnish with the sauteed potatoes and broccoli.