Preheat your oven to 400°F. In a large rimmed baking sheet, combine the butternut squash, nutmeg, smashed garlic, 1 tablespoon of olive oil, ½ teaspoon of kosher salt, and ¼ teaspoon of black pepper. Toss until everything is well coated and spread it out in a single layer. Bake for about 35 minutes, stirring once halfway through, until the squash is fork-tender.

While the squash is baking, whisk together 2 whole eggs, 8 egg yolks, and the remaining 1 tablespoon of olive oil in a medium bowl. Set the egg whites aside for another use (they can be frozen in an airtight container for up to a year).

In the bowl of a stand mixer fitted with a dough hook, add the flour. With the mixer on low speed, gradually pour in the egg mixture. Mix until mostly combined, about 30 to 40 seconds, but expect some loose flour at the bottom of the bowl. Continue mixing on low speed until the dough is smooth and all the flour is incorporated, which should take about 2 to 4 minutes. Stop occasionally to scrape the dough from the hook and press it into the loose flour.

Transfer the dough to a clean work surface and knead it by hand until it becomes elastic, which should take about 3 to 5 minutes. The dough should feel similar to modeling clay. If it’s too sticky, sprinkle in more flour, 1 teaspoon at a time; if it’s too dry and cracking, add a few drops of water.

Wrap the dough in plastic wrap and let it rest at room temperature for at least 30 minutes or up to 2 hours before rolling and shaping.

Once the squash is done roasting, transfer it to a large bowl and mash it gently with a fork or potato masher until smooth, which should take about 1 minute. Mix in the goat cheese, pecorino Romano, and whole milk until everything is well combined. You should have about a cup of filling. Set this aside, uncovered, until you’re ready to use it.

After the dough has rested, cut it into 4 equal pieces and cover them loosely with plastic wrap. Take one piece at a time and flatten it to about ¼-inch thickness, roughly 5 by 4 inches. Lightly dust it with flour, then roll it through a pasta machine set to the widest setting. Fold the dough into thirds (like a letter) and roll it through the widest setting again, with the folded edge first. Repeat this folding and rolling process one more time.

Continue rolling the pasta through the machine at progressively narrower settings, rolling it twice at each setting until the dough is thin enough to see the outline of your hand. Set the rolled dough aside on a lightly floured work surface and cover it with a clean kitchen towel. Repeat this process with the remaining pieces of dough.

Bring a large pot of salted water to a boil over medium-high heat. Take one sheet of dough and trim the edges to create a 24-inch-long rectangle. Cut the sheet in half lengthwise, making strips about 2 to 2 ¾ inches wide. Spoon 10 teaspoons of the filling down the center of one strip, spacing them about 2 fingers apart. Lightly brush water around the filling, then cover with the other strip of dough. Use your fingertips to smooth the dough and press out any air bubbles, sealing the edges and sides together.

Using a fluted or straight pastry cutter or a sharp knife, cut between the fillings to create 10 ravioli, each about 2 to 2 ½ inches square. Place the ravioli in a single layer on a large rimmed baking sheet dusted with flour and cover with a clean kitchen towel. Repeat this process with the remaining dough and filling.

In a large skillet over medium heat, melt the walnuts and butter, stirring occasionally for about 3 to 5 minutes. Add the sage and cook, stirring constantly, until the butter turns light brown and has a nutty aroma, which should take about 4 to 5 minutes. Remove the skillet from the heat.

Add half of the ravioli (20 pieces) to the boiling water and cook until they are al dente and float to the surface, which should take about 3 minutes. Using a slotted spoon or spider, transfer the ravioli directly to the brown butter in the skillet. Repeat this process with the remaining ravioli.

Return the skillet to medium heat, add the kale along with the remaining ½ teaspoon of salt and ¼ teaspoon of pepper. Cook while gently tossing until the kale is wilted and the ravioli are fully coated, which should take about 2 minutes. Serve garnished with additional pecorino Romano.