Heat the oven to 450°F with a rack in the middle position. Season the chicken on all sides with salt and pepper. In a 12-inch oven-safe skillet over medium-high, heat the oil until smoking. Add the chicken, skin down, and cook until fat is rendered and the skin is golden brown, about 5 minutes.

Transfer the chicken skin up to a plate. Pour off all but 1 tablespoon of fat from the skillet. Stir in the garlic and shallot and cook over medium, stirring occasionally, until light golden brown, about 1 minute. Add the vinegar and broth and bring to a simmer, scraping up any browned bits. Return the chicken to the skillet, skin up. Transfer to the oven and bake until the chicken reaches 175°F at the thickest part, or a skewer inserted into the thickest part meets no resistance, 12 to 15 minutes.

Transfer the chicken, skin up, to a deep platter and return the skillet to the stovetop (handle will be hot) over medium-high. Bring the sauce to a boil and cook until reduced to about 1 cup, 2 to 3 minutes. Stir in the olives, then taste and season with salt and pepper. Off heat, stir in half the rosemary, then spoon the sauce around the chicken. Top with Peppadews and the remaining rosemary.