Preheat the oven to 350 degrees F. Prepare a grill to medium-high heat.
In a small dish, combine the salt, chile powder, garlic powder and black pepper. Brush the chicken with the olive oil and sprinkle on all sides with the spice mixture.
Brush the grill with oil. Place the chicken skin-side down on the grill and cook for 5 minutes. Turn and cook an additional 5 minutes. Transfer the chicken to a baking dish in a single layer.
In a medium bowl, stir together the BBQ sauce and chicken broth. Pour the sauce over the chicken and cover the baking dish tightly with foil. Bake until the chicken is cooked through and fork-tender, an additional 40 minutes.