Heat the oven to 450°F with a rack in the lower-middle position. On a rimmed baking sheet, combine the squash, oil and 1 teaspoon each nutmeg and salt, tossing to coat. Distribute the squash in an even layer, then roast until lightly browned, about 20 minutes, stirring once about halfway through. Remove from the oven and sprinkle evenly with ½ cup Parmesan; stir, then redistribute in an even layer. Continue roasting until the cheese is browned and a skewer inserted into the squash meets no resistance, 15 to 20 minutes, rotating the baking sheet about halfway through.

Meanwhile, in a large pot, bring 4 quarts water to a boil. Add the pasta and 1 tablespoon salt, then cook, stirring occasionally, until just shy of al dente. Reserve 1½ cups of the cooking water, then drain and set aside.

In the same pot over medium-high, melt the butter. Add the sage and cook, stirring occasionally, until fragrant and the butter is golden brown, 1 to 2 minutes. Add the almonds, stirring just to coat. Add the pasta, the reserved cooking water, remaining ½ cup Parmesan, the vinegar, remaining ½ teaspoon nutmeg and ½ teaspoon salt. Cook, stirring occasionally, until the pasta is al dente and lightly sauced, 2 to 3 minutes.

Using a wide metal spatula, scrape up the roasted squash and any crusty bits of cheese and add to the pasta mixture; stir to combine. Off heat, taste and season with salt and pepper.