Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve about 1/2 cup of the pasta water, then drain.
While the pasta cooks, heat the olive oil in a large skillet over medium heat. Add the sliced chicken sausage and cook until browned and heated through, about 4–6 minutes.
Push the sausage to the side and add the minced garlic to the pan. Sauté for about 30 seconds until fragrant, then pour in the chicken broth to deglaze the pan, scraping up any browned bits.
Stir in the heavy cream and bring to a gentle simmer. Let it cook for 2–3 minutes to thicken slightly.
Add the spinach in batches, stirring until it wilts. If the sauce thickens too much, splash in a little of the reserved pasta water to loosen it.
Toss the drained pasta into the sauce, along with the parmesan cheese. Stir until everything is well coated and creamy. Season with salt, pepper, and red pepper flakes if you like a little heat.
Serve immediately, sharing portions into bowls or plates. A extra grating of parmesan on top never hurts.