Preparation

1. 1. Preheat oven to 400°F with a rack in the middle position. Melt 2 tablespoons of butter.
2. 2. Put the squash, shallot, and garlic cloves on baking sheet. Add the oil, melted butter, brown sugar, ¾ teaspoon salt, pepper, 4 sage leaves, and thyme and toss to coat. Spread into an even layer. Cover loosely with foil. Roast, stirring once or twice, until squash is very tender, about 45 to 50 minutes.
3. Uncover and let cool to room temperature.
4. 3. Meanwhile, insert the chopping blade into the work bowl of the food processor. Add the walnuts and pulse to finely chop. Remove and reserve. With the unit running on High, drop the Parmesan through the feed tube and process until finely grated. Remove and reserve.
5. 4. Prepare ravioli filling. Peel garlic cloves; discard skin and herbs. Put garlic, shallot, and about half the squash into the work bowl along with ¼ cup of the reserved Parmesan, mascarpone, egg yolk, zest and nutmeg and remaining ¾ teaspoon salt. Process on High until smooth, about 45 seconds, stopping to scrape down the sides of the bowl. Add remaining squash and pulse to combine. Reserve.
6. 5. Use a pasta maker to roll and cut the pasta dough to ravioli thickness. Alternatively, roll the dough out by hand with a rolling pin. Cut the dough into 6 equal pieces. Working with one piece at a time (keeping the rest of the dough covered), lightly flour both sides and roll lengthwise on a clean surface until you have a rectangle that is approximately 20 inches long by 6 inches wide. You should be able to see your hands through the pasta when you lift it. Transfer sheets to a semolina-dusted baking sheet and cover loosely with plastic wrap until ready to fill.
7. 6. Fill the ravioli. Place one sheet on a work surface lightly dusted with semolina. Determine the midway point of the sheet and brush the length of one half with egg wash. Starting 1-inch from the short edge, spoon mounds of filling, about 1 tablespoon each, down the center length of the egg-washed half, spacing at least 1½ inches apart. Fold the dry half of the sheet directly over the filled side. Press firmly in a circle around the mounds of filling to adhere, removing any air pockets with fingers. Use a pastry wheel or ravioli stamp to cut ravioli out evenly around the filling. Transfer ravioli tosemolina dusted baking sheet and cover loosely with plastic wrap until ready to cook. Continue with remaining pasta sheets and filling.
8. 7. Bring a large pot of salted water to a boil. In a large skillet, melt the remaining 4 tablespoons butter over medium-low heat. Continue to cook, swirling often, until foaming subsides and milk solids turn golden brown and smell nutty, about 3 to 4 minutes. Add the remaining sage leaves and remove from heat. In two batches, add ravioli to boiling water and cook until just tender, about 3 to 4 minutes, and transfer to skillet with a slotted spoon. Add ¼ cup pasta cooking liquid to the skillet and toss ravioli over medium-high heat, until sauce emulsifies and thickens to coat. Taste and season with salt and pepper, as needed.
9. 8. Divide ravioli and sage leaves among plates. Top with reserved walnuts and Parmesan.
10. Serve immediately.