In a large mixing bowl, combine the leftover mashed potatoes, flour, Parmesan cheese, egg, garlic powder, onion powder, salt, and black pepper. Mix until well combined. If using, fold in the chopped green onions.
Heat the vegetable oil in a large skillet over medium heat.
Scoop about 1/4 cup of the potato mixture and form it into a patty. Repeat with the remaining mixture.
Once the oil is hot, carefully place the patties in the skillet, making sure not to overcrowd the pan. Fry for about 3-4 minutes on each side or until golden brown and crispy.
Remove the pancakes from the skillet and place them on a paper towel-lined plate to drain excess oil.
Serve warm with sour cream, applesauce, or your favorite dipping sauce.