Bring a large pot of salted water to a boil. When the water boils, add the pasta and cook according to package instructions. Reserve 1 cup of pasta water, drain the pasta and return to the pot, drizzled with olive oil to prevent sticking.
In a large skillet, heat the olive oil over medium-high heat. When hot, add the sage leaves and a big pinch of salt and fry, stirring gently, until crisp, about 1-2 minutes. Transfer to a paper towel-lined plate to cool.
Add the onion to the skillet and cook over medium-low heat, stirring, until softened, about 5-7 minutes. Add the garlic, cayenne, and nutmeg and cook for 1-2 minutes, until fragrant. Add the grated squash and ½ cup water, cover, and cook for 10 minutes. Stir every 3 minutes or so.
Add 1 teaspoon salt, milk, and sherry vinegar to the squash and mash for 2 minutes. If you like, purée the squash with an immersion blender for a smoother consistency.
Add the squash mixture and ½ cup of pasta water to the cooked pasta, stirring to coat. Add more pasta water to reach your desired consistency and season to taste. Serve with fried sage leaves.