In a 6-inch-deep saucepan, heat 4 to 6 inches of oil to 375 degrees F.

Put the cooked and cooled potatoes through a food mill into a large bowl. Stir in 4 of the eggs, ham, and cheese and incorporate. Season with salt and pepper.

Pour flour onto a large plate and the breadcrumbs onto another large plate. Whisk the remaining 2 eggs with the milk and pour this mixture into another large plate.

Starting with a ball of potato mixture the size of an egg, form your croquettes using your hands (it helps to cover your hands with some of the flour to avoid a lot of sticking). They should look like fat cylinders. Dredge the croquette in flour, then the egg mixture, and finally, the breadcrumbs. Continue until all potato mixture is finished.

Carefully place 5 croquettes in the hot oil and fry until a deep golden brown, about 4 minutes. Remove and drain on paper towels. Continue until all croquettes are fried. Serve immediately.