Mix together the bread crumbs, parsley, salt and pepper. Set aside. In a small bowl, mix together the ham and cheese. Set aside. Pound the chicken cutlets thin. Lay the chicken cutlets out and divide the ham and cheese mixture between them, placing the mixture in the center. Roll up jelly roll fashion using toothpicks if you feel it is necessary. Dredge the rolls in flour. Coat with beaten egg, then roll in reserved crumb mixture covering well.
Sauce: In saucepan on the stove, melt butter and then add the flour. Stirring with a wisk until the mixture smells like peanut butter. Yes, put your nose down there and soon as the mixture smells like peanut butter, add the milk all at once. Whip until smooth, thick and creamy. Add spices, cheese and set aside. If you are making ahead, put a skim of butter on the top to keep the sauce from forming a skin.
In oil about an inch deep in a skillet, heat oil to 375 degrees. Brown the rolls on all sides being sure not to crowd in the skillet. Remove from skillet and drain on paper towels. Garnish and serve with the cheese sauce over top or on the side.