Gnocchi al Pesto (Potato Dumplings with Pesto)

Put potatoes into a 4-qt. pot of salted water; boil. Reduce heat to medium-low; simmer until potatoes are tender, 25 minutes. Drain; let cool. Peel potatoes; pass through medium plate of a food mill into a bowl. Add flour and eggs; using a fork, stir until dough forms. Transfer dough to a work surface; knead briefly to combine. Divide the dough into 6 portions. Roll each portion into a 1⁄2″-thick rope. Cut ropes into 1⁄2″-wide pieces; using the back of a fork, roll pieces along tines to imprint them with ridges.

Bring a large pot of salted water to a boil, and working in batches, add gnocchi and cook until just tender, about 5 minutes. Using a slotted spoon, transfer gnocchi to a large bowl and toss with pesto until evenly coated, adding a couple spoonfuls of cooking water, if needed, to create a smooth sauce. Transfer to a large serving platter or bowls and serve with more freshly grated parmesan.