In large bowl, mix mashed potatoes, egg and Parmesan cheese. Season with salt and pepper and mix well to combine.
In large nonstick skillet, heat vegetable oil over medium-high heat. In batches, drop heaping tablespoonfuls of potato mixture into skillet, pressing with spatula to flatten slightly.
Cook until golden brown on bottom, about 3 minutes. Repeat with remaining potato mixture. Serve warm with sour cream as condiment, if desired, and a side dish of chunky apple sauce.