Bring a large pot of salted water to boil for the pasta. You can start the pasta now but it will be done before the sauce so you'll need to set it aside. Otherwise, you can try to time the pasta to be done about 20 minutes after you start cooking the sausage.
In a large skillet, cook the garlic in the olive oil over medium heat for a minute, then add the sausage. Cook until no longer pink, stirring and breaking up the meat into small clumps. Season with the red pepper.
Add the broccolini, season with a bit more salt and add glug of olive oil, stirring to coat the broccolini. Add 1/2 cup of the pasta water, cover the pan and cook about 15 minutes, until the broccolini is tender.
Add another 1/2 cup of pasta water and bring to a boil. When the pasta is done, scoop it up and add it to the sauce, tossing to mix everything together.
Remove from the heat, mix in the cheese and serve.