In a 2-cup liquid measuring cup or small bowl, beat together the whole egg, egg yolks and 2 tablespoons water.

To make the dough in a food processor: Put the flour in a food processor. With the processor running, slowly stream in the egg mixture. Process until the dough leaves the sides of the bowl in large chunks, 1 to 2 minutes. If the dough feels dry and doesn’t hold together when pinched, add water, 1 teaspoon at a time, as needed. If the dough feels sticky, you will have the chance to knead in more flour after turning the dough out onto the counter.

To make the dough by hand: Put the flour in a large bowl and make a well in the center. Add the egg mixture; using a fork, stir the flour into the eggs, working from the outside edges into the center, until all of the flour is incorporated. Form into a rough ball. If the dough feels dry and doesn’t hold together when pinched, add water, 1 teaspoon at a time, as needed. If the dough feels sticky, you will have the chance to knead in more flour after turning the dough out onto the counter.

Lightly dust the counter with flour; if the dough feels wet and sticky, apply a heavier layer of flour to the work surface. Turn the dough out onto it and knead until smooth and shiny, 5 to 10 minutes. Press a finger into the surface of the dough; it should bounce back quickly, within 2 seconds. Cover the dough with a kitchen towel or plastic wrap and let rest at room temperature for 1 hour, or wrap tightly in plastic wrap and refrigerate up to 24 hours.

Line a rimmed baking sheet with a kitchen towel and lightly dust with flour. If the dough has been refrigerated, let it stand, still wrapped in plastic, at room temperature for about 15 minutes before proceeding.

Uncover or unwrap the dough and cut it into quarters. Set 3 pieces aside and cover with plastic wrap. Shape the remaining piece into a rough 4-by-6-inch rectangle. Using a pasta machine or a stand mixer fitted with a pasta attachment, roll the dough through several times, gradually reducing the thickness setting on the machine and dusting the dough with flour as needed, until it forms a long sheet about 1/16 inch thick. It’s important that the dough be of an even thickness. If the pasta sheet is longer than 14 inches, cut in half for shorter lengths.

Dust the dough sheet on both sides with flour, then accordion-fold it into thirds; set it on a cutting board. Using a chef’s knife and a decisive cutting motion (do not use a sawing action), cut the dough crosswise into ¼-inch to ½-inch strips. Unfold the pasta and transfer to the prepared baking sheet, gently separating the strands, then toss to lightly coat with flour; keep uncovered. Roll and cut the remaining dough in the same way. If not cooking right away, dust with additional flour and keep uncovered at room temperature for up to 1 hour, or cover with a kitchen towel and refrigerate for up to 12 hours.

In a large pot, bring 4 quarts water to a boil. Meanwhile, in a 12-inch skillet over medium, combine the butter, hazelnuts, sage, garlic and ½ teaspoon pepper. Cook, swirling the skillet and occasionally flipping the garlic, until the butter is amber and smells nutty, and the garlic and hazelnuts are golden brown, about 3 minutes. Set aside off heat. Remove and discard the garlic.

To the boiling water, add 1 tablespoon salt and the pasta, shaking off excess flour as you remove the noodles from the baking sheet. Cook, stirring, until al dente, 1 to 2 minutes. Reserve 1 cup of the cooking water, then drain the pasta. Return the skillet to medium-high. Add the pasta and ¾ cup of the reserved pasta water; cook, tossing to coat the noodles. Add the Parmesan and toss until the cheese is melted and the pasta is evenly coated, 1 to 2 minutes. Stir in the chives and lemon juice and zest, then remove from the heat. Add additional pasta cooking water 1 tablespoon at a time until the sauce is silky and clings to the noodles. Taste and season with salt. Serve sprinkled with additional pepper and Parmesan.