Preheat the oven to 200 degrees F.
In a bowl, mix the potatoes, egg, and milk together until combined. Add the flour, season with salt and pepper, and mix until smooth. Heat 2 teaspoons oil on a non-stick griddle over medium heat; pour about 1/3 cup of the batter onto the warm griddle, gently spread it to a circle about 4" wide, and cook, flipping once, until golden brown, 3-4 minutes per side. Transfer to the oven to keep warm. Repeat with remaining batter.
Heat 1 tablespoon of oil in a small skillet over medium-low heat. Add the onion and garlic and cook until softened, about 5 minutes. Add the red pepper and cook until softened, about 4 minutes more. Add the turkey, gravy, and corn and cook, stirring, until turkey is heated through. Season hash with salt and pepper, stir in the parsley, and keep warm over low heat.
Heat the remaining teaspoon of oil on the griddle over medium-high heat. Crack the eggs onto the griddle and cook until desired doneness. To serve, put a warm potato cake on each of 4 plates; top the cakes with a fried egg and divide the hash evenly among them.