Preheat the oven to 400°F.
Season the chicken halves on both sides with the salt and pepper. Set a 12-inch, cast-iron skillet over high heat and, when hot, add 1 tablespoon of the butter to the pan. Swirl the pan to evenly coat then lay the seasoned chicken halves, skin-side down, in the pan. Sear until golden, about 3 minutes then place the skillet in the oven. Cook until the chicken until nearly cooked through and the skin is crispy, about 17 minutes. Turn the chicken over and continue to cook, skin-side up, until the chicken is cooked through, 3 to 5 minutes longer.