Stir 1 egg in a large bowl to blend. Mix in the potatoes, then the flour. Using a generous 1/2 cup of potato mixture for each, form the potato mixture into 4 (4 1/2-inch) diameter pancakes.
Heat the oil in a large nonstick fry pan over medium heat. Fry the pancakes until they are golden brown and heated through, about 2 minutes per side. Transfer the pancakes to paper towels to drain. Keep the pancakes warm on a baking sheet in the oven. Pour off the excess oil from the pan.
Heat the pan over medium-low heat. Crack the remaining 4 eggs into the pan. Sprinkle with salt and pepper. Cook until the white is firm, about 2 minutes. Using a spatula, turn the eggs over and cook for 30 seconds longer.
Spoon the sauce onto 4 plates. Place a pancake atop the sauce. Top each pancake with a fried egg. Sprinkle with the Parmesan and serve.