In a large pot of boiling salted water, cook pasta until al dente, about 12 minutes.
Meanwhile, in a large skillet, heat oil over medium heat. Add onion and garlic and cook, stirring often, until softened, about 5 minutes. Add chicken broth and crushed red pepper and simmer, stirring occasionally, until broth is reduced by half and onion is tender, about 4 minutes. Stir in broccoli florets and cook over low heat until warmed through, about 3 minutes.
Just before draining pasta, pour out 1 1/2 cups cooking water and set it aside. Drain pasta thoroughly in a colander and then return it to pot. Remove garlic from broccoli skillet sauce; discard.
Add sauce, salt, and 1 to 1 1/2 cups pasta cooking liquid, as desired, to pasta to make a nice sauce. Toss well to coat pasta. Sprinkle with Parmesan and toss again. Serve with extra Parmesan.