Step 1

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.

Step 2

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.

Step 3

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.

Step 4

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.

Step 5

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.)

Step 6

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.