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 Tbs of the fat from the skillet.
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 wine and chicken stock to the skillet and stir to combine with vegetables.
Put the chicken back into the skillet. 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.
Remove the chicken from the skillet, 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 remove the skillet from the heat.
Transfer the chicken to wide soup plates; taste and season the sauce with salt and pepper if necessary and spoon it over the chicken. Sprinkle with the chopped parsley.