Heat oil in a large skillet over medium heat. Cook sausage until it is browned. Remove sausage from skillet and put skillet with sausage drippings to the side. Slice the sausage into ¼-inch thick pieces.
Bring a large pot of cold water to a boil over high heat. Add broccoli and cook, uncovered, until broccoli is tender. Immediately transfer broccoli to a colander (reserve boiling water) and rinse under cold running water. Chop into bite-sized pieces and set aside.
Add pasta to reserved boiling water and cook, about 12 minutes.
Meanwhile, heat the skillet with sausage drippings over medium heat. Add garlic and cook until it is golden brown, about 5 minutes. Add broccoli, red pepper flakes, and salt; increase heat to medium-high and cook, continually stirring, until broccoli is hot and completely tender, about 5 minutes. Add the sausage to the pan, mix well and cook until sausage is warmed, about 5 more minutes.
Drain pasta in a colander, reserving ¼ cup of the cooking liquid. Toss pasta and the reserved cooking liquid with the broccoli mixture.