Fill a large pot with salted water and bring to a boil over high heat. Add pasta and cook, stirring often, until al dente. Reserve 1 cup cooking water before draining; set aside. Pour pasta into a colander to drain; do not rinse.
Cut broccoli into florets. Trim and peel stalks and slice crosswise, 1/3-inch thick.
Cook sausage in a large skillet over medium heat, stirring and breaking meat into bite-sized pieces, until crispy and browned, about 5 minutes. Transfer to a plate.
Add broccoli, oil, garlic and red pepper flakes to pan and cook, stirring, until garlic is golden, about 3 minutes. Add chicken broth and simmer until broccoli is tender, about 3 minutes.
Transfer pasta, sausage, broccoli and broth to a large bowl, then toss well. (Add reserved cooking water from pasta as necessary for moisture.) Season with salt and pepper to taste.