In a large bowl, combine the mashed potatoes, cheese, stuffing, chopped turkey, egg. Using your hands, divide the mixture and then flatten into burgers about a 1 inch thick.
Heat 3 to 4 tablespoons of vegetable oil or butter in a large fry pan over medium heat. Make sure the bottom of the pan is thoroughly coated.
Fry the patties, in batches, until they're golden brown and crispy on both sides, around 4 or 5 minutes. Add more oil or butter to the pan as needed between batches. Do not overcrowd the pan and do not flip the burgers too soon or they won't develop a crisp crust and may fall apart. Transfer the burgers to a paper towel-lined plate and immediately sprinkle them with a dash of salt.
Serve the potato burgers topped with leftover gravy, ketchup or sour cream
NOTE: Leftover mashed potatoes vary in consistency depending on how much cream, milk or melted butter were added to the original batch. If the mashed potato burger mixture looks too dry and isn't holding together, add one more egg. If it looks too wet, add some flour, 1 tablespoon at a time, until the mixture is cohesive.