Melt the butter in a large skillet over medium heat. Season the chicken with salt and pepper and cook it in the skillet on the skin side to brown it and to render the fat from the skin, about 15 minutes. Turn the chicken over and cook on the bone side for about 5 minutes. Remove and reserve the chicken. Pour off all but 2 tablespoons of the fat from the pan. Add the onion, carrot, celery, and prosciutto or ham and sauté until the onion is browned and the vegetables are softened, 8 to 10 minutes. Add the vermouth and chicken stock to the pan and stir to combine with vegetables. Put the chicken back into the pan. Bring the liquid to a boil, reduce to a simmer, cover the skillet with a lid, and simmer the chicken in the sauce until cooked through, 20 to 25 minutes. Make the rice. In a pot, add rice and fill with water just to one inch above the rice. Cook over medium heat with lid on. Do not remove lid or stir the rice during the cooking process. When rice is done, fluff it gently. Remove the chicken from the pan, turn the heat to high and bring the liquid to a boil. Reduce the volume of liquid by half, about 5 minutes (or 6 to 7 minutes if you want a thicker sauce). Add the cream, stir to combine, and return the chicken to the pan. Simmer until the chicken is coated and hot. Sprinkle with the chopped parsley. Serve over pasta or a bed of rice.