A butcher will butterfly and pound the turkey breast for you.
Make the stuffing mix according to package and set aside. Place the turkey breast on a work surface, season the inner side with salt and pepper and cover with the stuffing mix spreading to cover. Roll the meat tightly and tie securely with string. At 2-3” intervals as well as the ends.
Heat the oil over medium heat in a pan that fits the roast snugly. As soon as the oil is hot, lightly salt and pepper the roast and add to pan. Cook until the turkey is lightly golden on all sides, 8-10 minutes.
Discard about ½ of the fat in the pan. Add the butter and milk, and bring the milk to a boil. Reduce the heat to low and cover pan. Cook 1 hour or until turkey breast reads 160ºF on a meat thermometer. Baste and turn the meat several times during cooking.
Remove the lid and cook, stirring until the milk is almost evaporated and only brown clusters of thickened sauce are left in pan.
Transfer turkey to a platter.
Tilt the pan and remove as much fat as possible. Raise the heat to high, and add the chicken broth, scraping the bits of the browned bits from bottom of pan and stirring in. Add the cream and the cheese and stir until smooth. Add the parsley and add salt and pepper as desired.
Slice the meat, and serve with the gravy.