Boil the potatoes in their skins until tender. Leave to cool slightly, then peel and press through a potato ricer while still hot. When cool transfer it to a large bowl and add the egg yolks, parmesan, mozzarella, pecorino, parsley, ham, salt, pepper and mix to combine.
Take a tablespoon of mixture and form it into a long fat finger shape.
Lightly whisk the egg whites in a shallow bowl.
Roll the croquettes in the flour, the egg whites and then the breadcrumbs. Repeat until all are complete.
Heat a deep frying pan with oil; make sure the oil is not smoking. Cook the croquettes in batches, do not over crowd the pan.
Serve as they are, and sprinkle with extra parmesan. They can also be eaten cold.