Bring a large pot of salted water to a boil. Add the pasta and cook according to the package instructions until it is al dente. Before draining the pasta, reserve about half a cup of the starchy pasta water to use later if needed.
While the pasta is cooking, heat the olive oil in a large skillet over medium heat. Add the minced garlic and a pinch of red pepper flakes. Cook for about 1 minute until the garlic is fragrant, taking care not to let it brown or burn.
Whisk the pumpkin puree, vegetable broth, heavy cream, and dried sage into the skillet. Stir constantly until the mixture is smooth and well combined. Let the sauce simmer gently for 3 to 5 minutes until it starts to thicken slightly.
Add the cooked pasta and the grated Parmesan cheese to the skillet. Toss everything together until every piece of pasta is evenly coated in the creamy pumpkin sauce. If the sauce feels too thick, add a tablespoon or two of the reserved pasta water to reach your desired consistency.
Taste the pasta and season with salt and black pepper as needed. Remove from the heat and serve immediately while the sauce is at its creamiest.