Preheat the oven to 450˚ F. Heat the olive oil in a large ovenproof skillet over medium heat. Season the chicken on both sides with salt and pepper and lay skin-side down in the skillet. Cook until the skin is well browned, 8 to 10 minutes. Transfer skin-side up to a plate. Pour off all but about 1 tablespoon drippings into a small bowl and reserve.
Add the olives, garlic and oregano to the skillet and cook until the garlic is softened, about 2 minutes. Pour in the chicken broth and 1 tablespoon lemon juice. Return the chicken skin-side up to the skillet. Transfer the skillet to the oven and roast until the chicken is no longer pink near the bone, 12 to 15 minutes.
Meanwhile, put the potatoes in a medium saucepan and cover with cold water; add a few big pinches of salt. Bring to a boil, then reduce the heat and simmer until the potatoes are tender, 10 to 12 minutes. Drain and return to the pan. Add 1 tablespoon of the reserved chicken drippings, the lemon zest, half the dill and a big pinch each of salt and pepper and toss.
Divide the chicken and potatoes among plates. Stir the remaining 1 tablespoon lemon juice into the sauce in the skillet; season with salt and pepper, then stir in the remaining dill and spoon over the chicken and potatoes. Top with the feta.