1 cup freshly grated Parmesan, plus more for serving
1/4 cup chopped fresh flat-leaf parsley (optional)
Cut the chicken breasts into 1-inch pieces. Season with salt and pepper. 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, but still a bit pink inside, about 4 minutes (the pieces don’t have to be browned on all sides; two sides is fine). Remove the chicken and set aside on a plate.
Do not clean the pan! Those brown bits on the bottom of the pan are going to add flavor to the sauce. Add the garlic to the pan and sauté over medium heat until you can smell it, 30 seconds.
Turn the heat to high, add the chicken broth, and scrape the bottom of the pan to loosen up all of those delicious caramelized bits. Bring to a simmer, lower to medium heat, and simmer for 5 minutes.
Add the pasta, stir well, and simmer until the pasta starts to soften, about 8 minutes.
Stir in the warm cream or half-and-half and the browned chicken with any juices that have accumulated on the plate. Cover and simmer, stirring occasionally, until the pasta is tender, most of the liquid has been absorbed, and the chicken is cooked through, about 4 minutes more.
Stir in the Parmesan until well incorporated, taste, and adjust the seasonings if needed. You can continue with the next step or see the Recipe Notes below for add-in suggestions.
Transfer the mixture to a serving bowl and sprinkle with the parsley, if desired. Serve hot and pass extra Parmesan at the table.