How to Make It
Preheat the oven to 425°. Rub the cut sides of the squash with 1 tablespoon of the olive oil and season with salt and pepper. Place the squash cut side down on a baking sheet and roast for 30 minutes, until tender. Using a fork, scrape the squash strands into bowls and keep warm.
Meanwhile, in a large pot, melt the butter in the remaining 2 tablespoons of oil. Add the onion and cook over moderate heat, stirring, until golden, 5 minutes. Add the mushrooms and garlic and cook, stirring, until the mushrooms are golden, 7 minutes. Add the broth and sage, season with salt and pepper and bring to a boil. Add the pasta and cook until al dente, 9 minutes. Discard the sage sprig and stir in the 1/4 cup of Parmigiano-Reggiano.
Ladle the soup over the squash and garnish with the chives. Serve with grated cheese.