Bring a large pot of water to a boil. Salt the water generously and add the pasta. Return the water to a boil and cook the pasta according to the package directions, but stop when it is very al dente, about 2 minutes before it's done. Reserve 1 cup of the cooking water and drain the pasta.
Meanwhile, heat the olive oil in a very large skillet over medium-high heat. Add the garlic, chicken, oregano, and salt and pepper and sauté until the chicken is just lightly browned on the outside, but not cooked through, about 3 minutes.
Add the chicken broth, 1/2 cup of the reserved pasta water, the broccoli florets, and pesto, if using. Cover the skillet and bring to a simmer. Add the drained almost-cooked pasta to the skillet, as well as the red-pepper flakes and butter and stir well. Reduce the heat to medium, cover the pot, and simmer until the pasta and broccoli are just tender, about 2 minutes. Add the lemon juice, if using, and the additional pasta cooking water if the amount of liquid seems too skimpy - you want it a little saucy. Stir in the Parmesan cheese, add additional salt and pepper if needed, and serve, passing additional Parmesan on the side.