Cook the fettuccine according to package directions until al dente. Before draining, reserve 1/2 cup of the starchy pasta water. Drain the pasta and set aside.
Season the chicken pieces lightly with salt and pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook until browned and cooked through, about 6-8 minutes. Remove the chicken from the skillet and set aside.
Reduce the heat to medium. Add the butter to the same skillet. Once melted, add the minced garlic and cook for about 1 minute until fragrant. Do not let the garlic brown.
Pour in the heavy cream. Bring the mixture to a gentle simmer, stirring occasionally. Let it simmer for 3-5 minutes until it slightly thickens.
Reduce the heat to low. Gradually whisk in the grated Parmesan cheese until the sauce is smooth and fully incorporated. Stir in the salt, pepper, and nutmeg, if using.
If the sauce seems too thick, slowly whisk in the reserved pasta water, one tablespoon at a time, until you reach your desired creamy consistency.
Return the cooked chicken to the skillet. Add the drained fettuccine pasta to the sauce. Toss everything together until the pasta is evenly coated.
Serve immediately, garnished with fresh parsley and extra Parmesan cheese.
Notes
Use freshly grated Parmesan cheese; pre-shredded cheese contains anti-caking agents that prevent the sauce from becoming smooth.
For a richer flavor, substitute half of the heavy cream with half-and-half, but watch the simmering time as it may require less reduction.
If you prefer a thicker sauce without adding more cheese, let the cream simmer uncovered for an extra minute or two before adding the cheese.