Boil the chopped potatoes in a pot of salted water until soft, then drain and mash them with a potato masher or fork.
Mix in the butter, egg yolk, Parmesan cheese, salt, and pepper until well combined.
Roll the mixture into small croquette shapes and place them on a plate or baking sheet.
Refrigerate the croquettes for at least 30 minutes to firm them up.
In separate bowls, place the flour and breadcrumbs.
Dip each potato croquette in the flour, shaking off any excess. Then dip in the beaten egg, allowing any excess to drip off. Finally, coat in breadcrumbs.
Heat enough vegetable oil in a deep pan or fryer to cover the croquettes.
Once the oil is hot (around 180°C), add a few croquettes at a time and fry them until golden brown, turning them occasionally for even cooking.
Remove the croquettes with a slotted spoon and place them on a paper towel-lined plate to drain any excess oil.
Serve the potato croquettes hot as a side dish or snack