How to Make It
In the bowl of a standing mixer fitted with the paddle, mix the flour with 1 teaspoon of salt. Add the egg yolks and mix at medium speed until the dough is blended and resembles wet sand, about 1 1/2 minutes. Transfer to a work surface and knead until a firm dough forms. Cover with plastic wrap and let stand at room temperature for 30 minutes.
Cut the pasta dough into 3 equal pieces and keep covered with plastic wrap. Working with one piece at a time, flatten the dough and run it through a pasta machine: Begin at the widest setting and work your way through consecutively narrower settings until you reach the thinnest one. Cut the sheet into 3 equal pieces. Run each piece through the spaghetti setting and transfer to a large rimmed baking sheet; toss the strands to keep them separate. Repeat with the remaining pasta dough.
In a large skillet, melt the butter. Add the sage leaves and cook over moderate heat, turning once, until crisp, about 3 minutes. Season the sage with salt and pepper. Using a slotted spoon, transfer the sage to a plate; reserve the butter.
In a large pot of boiling salted water, cook the pasta, stirring, until al dente, about 1 minute. Drain the pasta, reserving 1/4 cup of the pasta cooking water. Return the pasta to the pot, toss with the reserved butter and pasta cooking water and season with salt and pepper. Transfer to a large bowl and scatter the sage leaves and the 1/4 cup of cheese on top. Serve at once, passing additional cheese at the table.