Heat oil in a heavy-bottomed pot or dutch oven over medium heat. Add sausage and cook, breaking up with a spoon, until well browned and no longer pink inside. Transfer to a bowl.
Add onion to same pot, along with another splash of oil if needed. Sauté until translucent, then add garlic and cook a minute more. Add chicken broth, salt and squash and bring to a boil, scraping up any browned bits from the bottom of the pot. Cook, covered, until squash is very tender, about 15 minutes.
Meanwhile, add fettuccine to boiling water and cook 9 minutes or until al dente according to package directions. Reserve 1 cup of cooking water, then drain in a colander and return to the pasta pot.
Transfer squash mixture to blender along with cream and carefully process until very smooth (if your blender doesn't have steam vents, use an immersion blender instead, or else puree in two batches). Pour over pasta and toss, adding reserved cooking water as needed to thin it out. Add sausage and toss. Serve topped with black pepper and Parmigiano-Reggiano.