Cut the chicken breasts into 1-inch pieces. Season with salt and pepper.

Lightly steam or blanch the broccoli florets, then plunge them into a bowl of ice water to stop the cooking.

Melt the butter in a very large skillet over medium-high heat. Add the chicken, in batches if necessary, and sauté until nicely browned on the outside, and cooked through. As the chicken cooks, remove it from the pan to a plate and continue cooking all the chicken.

Do not clean the pan! The brown bits on the bottom are going to add flavor to the sauce. Add the garlic to the pan and sauté over medium heat, briefly, until you can smell it, about 30 seconds. Turn the heat to high, add the chicken broth, and scrape the bottom of the pan to loosen up all of those caramelized bits. Bring to a simmer, lower to medium heat, and simmer for 5 minutes. Add the pasta, stirring well, and cook until the pasta starts to soften, about 8 minutes. Stir in the warm cream and the chicken with any juices that have accumulated on the plate. Cover the pan and simmer, stirring occasionally, until the pasta is tender, much of the liquid has been absorbed leaving a creamy sauce, about 4 minutes more.

Stir in the Parmesan until well incorporated, taste, and adjust the seasonings.

Transfer the mixture to a serving bowl and sprinkle with the parsley to garnish. Serve hot and pass extra Parmesan at the table.