In a large pot of boiling salted water, cook the pasta, stirring often, until al dente. Drain (**this is very important) reserving 2 cups pasta cooking water.
Set a large skillet over medium-low heat and add 1-cup pasta water. Bring just to a low simmer, then whisk in 1 tablespoon of butter, one piece at a time, until melted.
Gradually add the cheese, whisking constantly, being sure it is melted and incorporated thoroughly before adding each addition.
Whisk in the additional 1-cup pasta water.
Add the pasta and mix to coat until noodles are covered in sauce.
(Note – if using cubed chicken, add it now and mix well.)
Taste and season with kosher salt and freshly ground black pepper, if desired.
Cook over low heat, about 3-5 minutes, just until the pasta absorbs most of the liquid. Stir often to avoid scorching the cheese-covered pasta.
To serve, place 1/4th of the pasta on a plate and add 1 sliced chicken breast.
Top with a heaping helping of grated Parmesan cheese and garnish with chopped fresh parsley or basil leaves.