Step 1Heat the oven to 350. Prick the potatoes with a fork. Place the potatoes on a baking sheet and bake them until they are soft, about 1 1/2 hours.
Step 2Immediately out of the oven, cut each potato in half and carefully scoop the potato out of the skin and into a fine-holed ricer. Pass the potatoes through the ricer and spread them out onto a clean work surface. Using a pastry scraper, chop the potatoes and season generously with white pepper.
Step 3Beat the egg yolk and drizzle it over the potatoes. Measure 1 1/4 cups of flour and sprinkle this over the potatoes. Cut the flour and egg into the potatoes until they resemble coarse crumbs. Bring the mixture together into a ball.
Step 4Sprinkle a scant one-fourth cup of flour onto the work surface. Spread the potato mixture over the flour, then top with another scant one-fourth cup of flour. Using the scraper and your hands, fold and press the dough until the flour is incorporated. Repeat. If the dough still feels tacky, repeat once more, this time covering the table and the dough each with 2 tablespoons of flour. You should fold the dough 8 to 10 times total.
Step 5Roll the dough into a compact log, smacking it on all sides for about a minute to condense it and remove any air bubbles. Dust the outside of the dough and your work surface with flour. Divide the dough into 8 pieces. Roll each section into a log about one-half-inch thick. Using a floured knife or pastry cutter, cut the dough into gnocchi about 1 inch long. This will make about 300 gnocchi.
Step 6Bring a large pot of heavily salted water to a boil (the seasoning in the water will season the gnocchi). Working in two or three batches, drop the gnocchi into the water and cook, stirring occasionally until they float, 2 to 3 minutes. Remove the gnocchi from the water immediately and refresh them in ice water. Drain them and refrigerate until ready to serve (gnocchi should not be cooked more than about 6 hours before they are served).
Step 7When ready to serve, bring one-half-inch of water to a simmer in a medium saucepan over medium heat. Reduce the heat to medium-low and whisk in the butter a piece at a time. Toss the butter with the gnocchi and stir gently to coat them with sauce as they warm (if needed, stir the gnocchi and sauce in a large skillet or saute pan over low heat to warm the gnocchi thoroughly). Season with salt and white pepper and serve topped with grated cheese.