Bring 8 quarts of water to a boil. Cook the potatoes in the boiling water until easily pierced with a paring knife, about 15 minutes. Drain the potatoes and, while still hot, pass them through a food mill. Immediately add the egg yolks, prosciutto, butter, Parmigiano-Reggiano, and nutmeg and mix well. Season with salt and pepper.
With moist hands, divide the mixture into golf ball-sized portions and form disks of a 2-inch diameter.
On three separate plates, place the flour, egg whites and bread crumbs. Dredge each disk through each of the plates, starting with the flour, then the egg whites, then the bread crumbs. In a heavy-bottomed skillet, heat the oil until almost smoking. Cook the disks in the oil, working in batches to avoid overcrowding the pan, until disks are deep golden brown. Remove disks to a plate lined with paper towels and serve. They can be reheated from room temperature, but should not be refrigerated.