In a large bowl, combine the mashed potatoes, egg, flour, Parmesan cheese if using, green onions, salt, and black pepper. Mix until the mixture is well combined and holds together.
Form the potato mixture into 8 small patties, about 1/2 inch thick. If the mixture is too sticky, lightly dust your hands with flour.
Heat the butter and olive oil in a large skillet over medium heat until shimmering.
Place the patties in the skillet and cook for 3 to 4 minutes per side, or until golden brown and crisp. Cook in batches if needed to avoid overcrowding.
Transfer the pancakes to a plate lined with paper towels to drain briefly. Serve warm with sour cream, chives, or your preferred sauce.