Add the cold mashed potatoes, cheddar cheese, mozzarella cheese, Parmesan cheese, parsley, egg, flour, garlic powder, onion powder, smoked paprika, salt, and black pepper to a large bowl.
Mix until evenly combined. If the potato mixture is too soft to roll, add more flour, 1 tablespoon at a time, until it holds its shape.
Scoop about 2 tablespoons of the mixture and roll it into a ball about 1½ inches wide. Place it on a parchment-lined baking sheet or plate. Repeat until all the mixture has been used.
Refrigerate the potato balls for at least 20 minutes to firm them up.
Prepare three shallow bowls. Add flour to the first bowl. Add beaten eggs to the second bowl. Add panko breadcrumbs, salt, and smoked paprika to the third bowl.
Coat each chilled potato ball in flour, shaking off excess. Dip it into the beaten egg, then roll it in the panko mixture, pressing lightly to help the breadcrumbs stick.
Pour about 2 inches of oil into a deep skillet, saucepan, or Dutch oven. Heat the oil to 350°F / 175°C.
Fry the potato balls in small batches for 3 to 4 minutes, turning gently, until golden brown and crispy. Do not overcrowd the pan.
Transfer the cooked potato balls to a wire rack or paper towel-lined plate. Sprinkle lightly with salt while hot.
Serve immediately with garlic aioli, ranch, ketchup, sour cream, marinara, spicy mayo, or your favorite dipping sauce.