Bring a large pot of water to a boil. When water boils, add a generous handful of salt, add pasta and broccoli and cook according to pasta instructions for al dente. When pasta is almost done cooking, reserve about ½ cup of the pasta water and set aside. Drain the pasta and broccoli.
Meanwhile, heat olive oil in a large skillet over medium-high heat. Add the garlic and chicken and cook, stirring occasionally until the chicken is browned, about 5 minutes.
Stir in the chicken broth and wine, if using, and bring to a boil, stirring occasionally.
Add the pasta and broccoli to the skillet, along with the grated cheese, salt and pepper and mix well.
If the mixture seems dry, add the reserved pasta water, a little at a time, using only as much as needed.
Serve immediately pasta bowls with additional grated cheese and red pepper flakes on the side