Boil the diced potatoes in salted water until tender. Drain and mash until smooth.
Stir in Parmesan, egg, salt, and pepper. Let cool slightly.
Take a spoonful of mashed potato, flatten it in your hand, and add a bit of mozzarella in the center. Shape into a ball or oval to enclose the cheese.
Roll each croquette in flour, dip in beaten egg, and coat with breadcrumbs.
Heat oil in a deep skillet or pot to 350°F (175°C). Fry croquettes in batches until golden brown, about 2-3 minutes per side.
Drain on paper towels and serve hot.