Preheat the oven to 350°. In a large cast-iron skillet, heat the oil. Season the chicken with salt and pepper, cook over moderate heat until browned all over, about 10 minutes total. Transfer the chicken to a plate. Pour off all but 1 tablespoon of the oil in the skillet. Arrange the potato slices in the pan, overlapping them slightly. Set the chicken skin side down on top of the potatoes. Scatter the prunes around the chicken. Mix the sour cream into chicken broth and pour the broth mixture and mushrooms on top. Roast the chicken for 20 minutes. Reduce the oven temperature to 300° and roast for about 45 minutes longer, until the potatoes are tender and the chicken is cooked through.
Preheat the broiler and arrange the rack 6 inches from the heat. Turn the chicken skin side up and broil until
the skin is crisp, about 8 minutes.