In a large bowl, stir together mashed potatoes, egg, and Parmesan. Season with salt and pepper and mix well to combine (mixture will look wet). In a large nonstick skillet, heat vegetable oil over medium-high.
In batches, drop heaping spoonfuls of potato mixture into skillet, pressing with a spatula to flatten slightly. Cook until golden brown on bottom, about 3 minutes. Gently flip and cook until golden brown on other side, 3 minutes. Repeat with remaining potato mixture and more oil.