Toss the cubed squash with the olive oil in a large bowl and season with ½ teaspoon of the salt and pepper. Transfer the mixture to a rimmed baking sheet in a single layer. Roast until tender and caramelized, about 30 minutes, stirring once halfway through and adding the garlic at the halfway point.

Remove the squash from the oven and let cool for 10 minutes. Transfer the squash and garlic to a blender or food processor. Add the remaining salt, cream, broth, cheese, and nutmeg, and process until smooth, about 1 minute.

While the squash is baking, bring a large pot of salted water to a boil. Add the fettuccine and cook until al dente, about 8 minutes or according to package instructions. Reserve 1/2 cup of the pasta water, then drain the pasta. Return the pasta to the pot.

Add the Alfredo sauce to the pasta and toss over low heat. If needed, add some of the reserved pasta water, a tablespoon at a time, to loosen sauce and evenly coat the pasta. Sprinkle with the chives and more Parmesan and serve immediately.