In a bowl, mix together the mashed potatoes, beaten egg, cheese, and salt. If your mashed potatoes are a too thin add flour 1 TBSP a time until they are able to hold their shape when cooking.
Over medium heat, add enough oil (1/4-1/2 cup) to a large skillet. You need the oil to be deep enough to cover at least half of the Potato Patty.
With an Ice cream scoop, take a large scoop of the potato mixture, then flatten into a patty. You should get about 12 patties from 2 1/2 cups of potatoes.
Place formed patties in the heat oil in the skillet. Cook on one side until golden brown about 2 minutes. Flip the patty to cook the other side, about another 2 minutes until golden brown and heated through.
Remove cooked patties from the skillet with a spatula and place on a paper towel to drain. Sprinkle with salt immediately.
Serve with sour cream and chopped chives, a splash of hot sauce or salsa. Delicious served on their own too!