Make the pasta. Keep the flour in the bowl or turn onto a work surface. Make a wide well in the center with your fist, forming a wall of flour tall enough on all sides to contain the liquid.
Using a fork, gradually incorporate a portion of the flour, a couple of teaspoons or so at a time, from the inner rim of the well into the liquid in a circular motion until a thick, custardy batter forms. Incorporate the remaining flour by pulling, folding, and pressing the flour into the center until it becomes a shaggy ball. Knead vigorously. After about 5 minutes, cover the dough tightly in plastic wrap. Let rest for 5-10 minutes. Then continue to knead the dough until it's smooth and firm.
Make the filling by combining cheeses and a generous pinch of salt and pepper to the bowl of a food processor. Pulse until very smooth, 30 to 60 seconds, scraping down the side of the bowl as needed.3. Make the filling by combining cheeses and a generous pinch of salt and pepper to the bowl of a food processor. Pulse until very smooth, 30 to 60 seconds, scraping down the side of the bowl as needed.
Assemble pasta. Use a pasta machine to roll the dough into very thin pasta sheets. Trim into 6-inch squares. Use a piping bag to pipe the filling "nest" that holds the yolk in place (a large zip-top bag with a corner snipped off will work in a pinch). Cover and seal around the filling. Trim the edges using a pasta wheel. Place on a sheet pan dusted in semolina.
In a large sauté pan or skillet, melt the butter over medium heat. Add the sage and fry until crisp and dark around the edges, swirling the pan often to coat the leaves in the butter, about 3 minutes. Remove from the heat and transfer the sage to a paper towel–lined plate.
Cook your pasta of choice to your liking. When the pasta is almost done, return the butter to medium heat. Scoop out ¾ cup pasta cooking water and add it to the butter. Whisk or stir the sauce constantly until emulsified.
Simmer the sauce until slightly thickened, about 2 minutes. Transfer the pasta to a serving platter or bowl and pour the sauce on top, making sure all the pasta is coated.
Serve topped with grated Parmigiano-Reggiano and the fried sage.