Bring a pot of salted water to a boil. Cook the fettuccine according to the package directions. Reserve 2 cups of the cooking water, then drain the pasta and rinse with hot water.
Heat the butter in a large skillet over medium heat. Add the cream and ½ cup of the pasta water. Bring to a simmer. Add the fettuccine, chicken, and 1¼ cups Parmesan cheese. Toss to coat. Season to taste with salt and pepper. If the pasta is dry, add more of the pasta cooking water, a little at a time. Serve immediately with additional Parmesan sprinkled on top.