If using freshly made mashed potatoes, spread them on a plate and chill for 10 to 15 minutes so they firm up a bit. Leftovers work great and often need no chilling.
In a mixing bowl, combine the mashed potatoes, egg, grated cheese if using, chives, salt, and pepper. Mix until well combined.
Add 1/2 cup flour and stir. The mixture should hold together when shaped. If it's too wet, add up to 2 tablespoons more flour. If too dry, add a splash of milk or another egg yolk.
Shape the mixture into 8 evenly sized cakes, about 3 inches across and 1/2 inch thick. Lightly dust each cake with a little flour, or for extra crispiness, press both sides into panko breadcrumbs.
Heat a large nonstick or cast-iron skillet over medium heat and add the oil. When the oil is shimmering but not smoking, add the potato cakes in a single layer without crowding the pan.
Cook for 3 to 4 minutes on the first side until golden brown and crisp. Flip carefully and cook another 3 to 4 minutes until the other side is evenly browned.
Transfer to a paper towel-lined plate to drain briefly. Serve warm with sour cream, applesauce, or a quick salad.