Bring a large pot of salted water to a boil and cook pasta according to package directions for al dente.
Add the broccoli florets to the boiling pasta water during the final 3 minutes of the pasta's cooking time.
Reserve 1/2 cup of the starchy pasta water, then drain the pasta and broccoli together.
In the same pot or a large skillet, heat the olive oil over medium heat.
Sauté the minced garlic and red pepper flakes for 60 seconds until fragrant but not browned.
Return the pasta, broccoli, and shredded chicken to the pot.
Pour in the reserved pasta water and add the Parmesan cheese.
Toss vigorously over low heat until the cheese and water emulsify into a light sauce that coats the pasta.
Season with salt and black pepper to taste and serve immediately.