In a large bowl, stir together 2 cups mashed potatoes, 1 large egg, and 1/4 cup parmesan cheese. Season with coarse salt and ground pepper and mix well to combine. In a large nonstick skillet, heat 1 tsp oil over medium-high heat. 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, about 3 minutes. Repeat with remaining potato mixture and more oil.
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