Preheat the oven to 400°F. To make the filling, cut the squash in half lengthwise and scoop out the strings and seeds. Put the squash halves in a baking dish, cut side up. Cover and bake until tender when pierced, about 1 hour. Cool, then scrape the flesh away from the skins. Puree the flesh in a food processor until smooth.

Measure 1 cup squash puree into a bowl. (Discard any remaining puree or reserve for another use.) Add the cheese, bread crumbs, and nutmeg and salt to taste. Stir in the egg yolk.

Roll the pasta on the pasta machine into sheets as thin as you are comfortable working with. (I roll it to number 7, two steps thinner than I do for fettuccine.) Lay the sheets on clean dish towels as you make them.

Prepare the ravioli while the pasta is still fresh and somewhat moist. On a work surface lightly dusted with semolina, lay 1 sheet flat. Dot the sheet with small mounds of filling in two parallel rows, using a scant 1 teaspoon filling for each mound. The mounds should be about 1 inch apart and the rows about 2 inches apart. Keep the mounds far enough from the edges of the dough to allow room to seal the ravioli. With a pastry brush dipped in cold water, lightly moisten the edges of the pasta dough and moisten a strip between the rows.

Top with another sheet of pasta dough (four hands are helpful here, but not essential), carefully aligning the edges. Press between the mounds and between the rows to seal the ravioli. Use a fluted pastry wheel to cut between the ravioli. Transfer them to semolina-dusted trays or dish towels to rest while you prepare the sauce. If you are not going to cook the ravioli immediately, you need to turn them every 20 minutes or so to prevent them from sticking to the trays or towels.

Bring two large pots of salted water to a boil over high heat.

Meanwhile, prepare the sage leaves: Melt 2 tablespoons of the butter with the olive oil in a large skillet over moderately low heat. Add the sage leaves and cook slowly, turning them occasionally, until they crisp, 5 to 7 minutes. Transfer them to paper towels to drain and sprinkle with salt.

Pour the fat from the pan into a small cup and let the dark particles settle to the bottom, then pour the clear fat back into the skillet. Add the remaining 6 tablespoons butter and melt over low heat. Season to taste with salt and pepper.

Add the ravioli to the boiling water, dividing them between the two pots. Cook until al dente, 3 to 4 minutes, depending on how dry they are and how thin your pasta is. Stir them often so they don’t stick to each other. Check by lifting one out with a strainer and cutting off a bit of the edge to taste. When the ravioli are done, lift them out of the pot a few at a time with a skimmer or strainer, letting excess water drip back into the pot. Divide them among warmed bowls, drizzling each portion with the melted butter and sprinkling with some of the cheese and sage leaves. Serve immediately.