To make the pesto, in a food processor, grind the pistachios, cheese and ¾ teaspoon each salt and pepper until they resemble coarse sand, 10 to 20 seconds. Add the arugula, oil and lemon juice; process until almost completely smooth, about 20 seconds. Transfer half (about ¾ cup) to a large bowl; set aside. (See headnote for how to store the remainder).

To make the gnocchi, in a large bowl, stir the potato flakes and water. Let stand until completely cooled. Mix in the eggs, flour and 1 teaspoon salt until a dough forms. Lightly dust the counter with flour and turn the dough out onto it. Gently knead until smooth.

Dust a rimmed baking sheet with flour. Divide the dough into 4 pieces and cover with a kitchen towel. Roll one piece into a rope 18 inches long and ¾ inch in diameter. Cut into ½-inch pieces and lightly dust with flour. Dip the back of the tines of a fork into flour, then gently press into each piece to create a ridged surface. Transfer the gnocchi to the baking sheet, spacing them so they don’t touch. Repeat with the remaining dough.

In a large pot, boil 4 quarts water. Add 1 tablespoon salt and half the gnocchi. Return to a boil, stirring once or twice; cook for 2 minutes (the gnocchi will float before they are done). Using a slotted spoon, transfer to the bowl with the pesto. Return the water to a boil, then repeat with the remaining gnocchi. Reserve ¼ cup of the cooking water.

Toss to coat the gnocchi with pesto, adding reserved cooking water 1 to 2 tablespoons at a time until the sauce clings. Taste and season with salt and pepper. Serve sprinkled with additional cheese.