Boil the pasta. Bring a large pot of salted water to a boil.
Cook the fettuccine until just al dente. Reserve about 1 cup of the starchy pasta water, then drain and set aside.
Season the chicken. Pat the chicken dry and season with salt and black pepper. Thin strips cook quickly and stay tender.
Sear the chicken. Heat olive oil in a large skillet over medium-high heat.
Add chicken and cook for 4–6 minutes, stirring occasionally, until lightly browned and cooked through. Transfer to a plate and tent with foil.
Start the sauce base. In the same skillet, reduce heat to medium. Add butter and let it melt.
Stir in the minced garlic and cook for 30–60 seconds until fragrant, not browned.
Add cream and milk. Pour in the heavy cream and milk. Stir and bring to a gentle simmer. Do not boil hard; keep it soft and steady.
Stir in Parmesan. Lower the heat.
Gradually add the grated Parmesan, stirring constantly until smooth and creamy. If the sauce looks too thick, loosen with a splash of pasta water. Add nutmeg if using.
Combine pasta and chicken. Add the cooked pasta to the sauce and toss to coat.
Return the chicken and any juices to the skillet. Warm everything together for 1–2 minutes, adding more pasta water as needed for a silky, clinging sauce.
Taste and adjust. Season with salt and black pepper. The Parmesan is salty, so taste before adding more.
Serve. Top with extra Parmesan and chopped parsley.
Serve immediately while the sauce is at its creamiest.