2 lb. medium or large red potatoes (about 5 medium), scrubbed and cut into large chunks, 1-1/2 to 2 inches
Kosher salt and freshly ground black pepper
1/4 cup olive oil
2 cloves garlic, minced
1/4 cup finely chopped fresh basil leaves
3/4 cup freshly grated Parmigiano Reggiano
Put the potatoes and 1 tsp. salt in a large saucepan or pot and fill with enough cold water to cover the potatoes by about 1 inch. Bring to a boil over high heat. Lower the heat to maintain a steady simmer, cover the pot partially, and cook until the potatoes are quite tender when tested with a fork, 15 to 20 minutes. Reserve some of the cooking water and then drain the potatoes. Dump them back in the pot and set over medium heat.
Dry the potatoes by shaking the pan and stirring until most of the moisture has steamed off, about 2 minutes. Remove the pan from the heat. Using the side of a large metal spoon, cut through the skins and flesh of the potatoes, reducing the chunks to a very coarse mash.
Stir in the olive oil and garlic. Add up to a couple of tablespoons of the cooking water to loosen the mash if necessary. Stir in the basil and cheese. Season to taste with salt and pepper and serve immediately.
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