In a large bowl, mix together the leftover mashed potatoes, grated cheese, chopped green onions, all-purpose flour, egg, salt, and pepper. Stir until everything is well incorporated, creating a thick and flavorful mixture.

Using your hands, form the mixture into small, even cakes about 2 inches wide. Make sure they are well-formed to prevent them from falling apart while cooking.

Add a generous amount of oil to a skillet and heat it over medium heat. You want the oil to shimmer before adding the cakes, which indicates it's ready for frying.

Carefully place your potato cakes in the hot oil, ensuring they aren’t overcrowded. Cook until they are golden brown on one side, roughly 4-5 minutes, then flip and fry until the other side is golden and crispy.

Remove the cakes from the skillet and allow them to drain on paper towels to absorb any excess oil. Serve hot for the best flavor and texture!