Preheat the oven to 425 degrees . In a small bowl, whisk the tapenade with 3 tbsp. oil.
In a cast-iron skillet, heat the remaining 1 tbsp. oil over medium-high. Season the chicken with salt and pepper. Cook, skin side down, until golden, 8 minutes. Transfer to a plate; pour off all but 3 tbsp. skillet drippings.
In the skillet, add the potatoes, shallots, garlic and poultry seasoning; season. Cook over medium-high, stirring often, until the potatoes start to brown, about 5 minutes. Arrange the chicken on top, skin side up. Brush the chicken with half the tapenade mixture.
Transfer to the oven and roast, brushing the chicken with the remaining tapenade mixture halfway through, until the chicken is cooked through and the potatoes are tender, 20 to 25 minutes. Divide among plates; drizzle with the skillet juices.