Step 1Heat the oven to 350 degrees. Place the walnuts on a baking sheet and toast until brown and fragrant, about 8 minutes. Set aside.
Step 2Put the potatoes, whole and unpeeled, in a large pot with enough cold water to cover them by 2 inches. Bring to a boil, and cook until they're easily pierced with a knife, about 20 to 25 minutes. Don't overcook or let the skins burst.
Step 3In a food processor, place the garlic, one-fourth teaspoon salt, the walnuts, lemon zest, Parmesan and parsley, and puree until smooth. Spoon the pesto mixture into a medium bowl and gradually stir in the walnut oil. Mix until combined and reserve.
Step 4When the potatoes are done, drain them and peel them carefully with a paring knife while they're still hot; use a kitchen towel to hold them. Put the peeled potatoes into the tamis, held over a large bowl. Using a bowl scraper, break the potatoes apart and scrape them through the tamis. Spread the potatoes into a thin layer on a baking sheet or tray, sprinkle the remaining three-fourths teaspoon salt over them, and let them cool and dry for at least 20 minutes and up to 2 hours.
Step 5To mix the dough, pile the dried potatoes into a large, loose mound on a board or work surface. Make a small well in the pile of potatoes, pour in the beaten eggs, then sprinkle 1 cup of the flour and the minced sage on top. Using your hands or the bowl scraper, work in the eggs, mixing and moistening the flour and potatoes. Gather into a single mass, and knead for several minutes, scraping in sticky bits from the board and your hands. Incorporate additional flour in small amounts, only as needed, until the dough is smooth, soft and only slightly sticky. Avoid adding too much flour, which will make the gnocchi heavy and dry. Cover the dough with a towel and form the gnocchi as soon as possible. Meanwhile, bring 8 quarts of water with 2 tablespoons of salt to a rolling boil.
Step 6To shape the gnocchi, cut the finished dough into three or four pieces. Dust the work surface and your hands with flour. Roll one piece under your hands into a thick cylinder, and gradually stretch it into a long rope, about two-thirds-inch thick. With a sharp knife or the bench scraper, slice the rope crosswise into half-inch lengths; sprinkle the pieces with flour.
Step 7To form the gnocchi, use a fork or a gnocchi paddle. Hold the fork or paddle with the tines (or grooves, if using a paddle) at an angle against your work surface. Place one of the cut sides of a piece of dough against the tines. With your lightly floured thumb, press into the dough, and at the same time push it off the end of the fork or paddle and onto a floured board. It will be hollow and curved where you pressed it, and ridged on the side that rolled off the tool. Press and roll the other cut pieces, dust them with flour, and set in a single layer on a floured tray, not touching. (Gnocchi should be cooked, or frozen, as soon as they are all shaped.)
Step 8When the water is at a rolling boil, brush off the excess flour from a large handful of gnocchi and drop them into the pot. Stir, cover the pot, and allow the water to return to a boil over high heat. As the gnocchi come to the surface, turn and stir them occasionally so that they cook evenly and don't stick to one another. Boil for a total of about 6 minutes, until cooked through. Remove with a slotted spoon or strainer. Repeat for the rest of the gnocchi.
Step 9While the gnocchi are cooking, spoon about one-fourth cup of pesto into a medium bowl and add a few tablespoons of the water you are cooking the gnocchi with into the bowl. Stir the pesto and water to combine. When the gnocchi are done, lift them out of the water with a slotted spoon and drop them into the bowl with the pesto. Stir gently to combine, then spoon the gnocchi into a soup plate. Sprinkle a little of the extra Parmesan on top, grind a little black pepper over it, adjust the seasoning and garnish with a few sage flowers. Serve immediately.