We've experimented with many dishes from Naples, and these potato croquettes are a classic Neapolitan fritto. Andrew adapted the recipe, making it his own, by adding provolone piccante, an aged cheese with pronounced sharpness that also contributes some moisture to the croquettes. For a good part of her early life, our daughter, Prue, basically subsisted on these: They would arrive at the table piping hot, we'd cut them in half to release some of the steam, and Prue would wait expectantly for her nibbles of the creamy croquettes as Andrew and I said, "Hot-hot-hot." Kids love these unanimously and so do adults. At the restaurant, we shower the croquettes with very finely grated Parmigiano-Reggiano. It's not necessary, but the effect is very pretty, like a snow-covered mountain. They're just a wonderfully simple yet polished little bite, and they'd make a superior snack for a cocktail party.
*Note: If you can’t track down provolone piccante, use a combination of cheeses — about 25 percent Pecorino and 75 percent caciocavallo.
In a large pot, combine the potatoes, water, and 2 tablespoons salt. Bring to a boil and boil until the potatoes are tender, 20-25 minutes. Drain.
Place the butter and provolone in a large bowl. Pass the hot potatoes through the medium-sized plate of a food mill into the bowl. Season with ¼ teaspoon salt and ¼ teaspoon pepper. Stir in 1 of the eggs.
Spread the batter in an even layer on a rimmed baking sheet and cool completely. Cover tightly with plastic wrap and chill for at least 6 hours and as long as overnight.
To make the croquettes, place the flour, remaining eggs, and the breadcrumbs in 3 separate wide shallow bowls. Lightly beat the eggs and roll the cold potato mixture into tablespoon-sized balls.
Dip each ball in the flour, turning to coat, then in the eggs, and then coat evenly with breadcrumbs. (At this point, the croquettes can be refrigerated on a baking sheet tightly covered with plastic wrap for up to 3 days; bring to room temperature before frying.)
In a deep-fryer or heavy pot, heat at least 2 inches of oil to 360 degrees and line a plate with paper towels. Working in batches, fry the croquettes until golden and crisp, 2-3 minutes. Transfer to the plate to drain. Toss with salt and pepper, to taste, and the Parmigiano-Reggiano while still warm.