Add enough cold water over the potatoes in a medium saucepan to cover them by 3 fingers.
Bring the water to a boil over high heat, then lower the heat slightly to a gentle boil.
Cook until tender when poked with a wooden skewer or the tip of a paring knife, 20-35 minutes, depending on the size of the potatoes.
Drain the potatoes and let them stand until cool enough to handle.
Scrape off the skins with a knife and pass the potatoes through a ricer or force them through a coarse sieve back into an empty pot.
Stir in the butter; et the pot over medium heat, and stir until the puree is smooth.
Remove from heat and season with salt, white pepper, and a touch of nutmeg.
Beat one of the eggs and stir it, along with the prosciutto and parsley, into the potato mixture until all the ingredients are evenly distributed; cool to room temperature.
Using about 2 tablespoons of the potato mixture for each, form cylindrical croquettes about 2 inches long.
Refrigerate the croquettes, uncovered, until firm, about 20 minutes.
They can also be covered and refrigerated for up to 1 day before continuing.
Whisk the remaining two eggs well in a wide bowl.
Spread the breadcrumbs and flour out on separate plates or sheets of waxed paper.
Working with a few at a time, dredge the croquettes in flour to coat them lightly and tap off the excess.
Dip the floured croquette into the egg, turning well to coat all sides evenly.
Let the excess egg drip back into the bowl, then lay the croquettes in the bread crumbs; turn to coat all sides and ends well with the bread crumbs, pressing very gently with your hands until the crumbs adhere well to the croquettes.
Heat the vegetable oil in a heavy, medium skillet over medium heat to 350°.
When the oil reaches temperature, slip half the croquettes into the oil.
Fry carefully, turning them as necessary, until golden on all sides, about 4 minutes.
Adjust heat under the pan as the croquettes cook so they cook evenly.
Remove the croquettes with a slotted spoon and drain them on a paper-towel lined baking sheet.
Repeat with the remaining croquettes.
Fried croquettes can be kept warm on a baking sheet in a 200° oven for up to 20-30 minutes.