Cook the pasta. Bring a large pot of salted water to a boil. Cook pasta until just al dente. Reserve 1 cup pasta water, then drain.
Cook the chicken. Pat chicken dry, season with salt and pepper. Heat olive oil (and butter, if using) in a large skillet over medium-high heat. Cook chicken 4 to 6 minutes per side (depending on thickness) until cooked through. Rest 5 minutes, then slice.
Start the sauce. In the same skillet, melt butter over medium heat. Add garlic and cook 30 to 60 seconds until fragrant.
Make the roux. Whisk in flour and cook 1 minute, stirring constantly.
Add dairy. Slowly whisk in half and half until smooth. Add salt, pepper, and nutmeg. Simmer gently 2 to 4 minutes until slightly thickened (do not boil hard).
Add cheese (off heat). Turn off the heat. Whisk in Parmesan until melted and smooth.
Toss and finish. Add cooked pasta and toss to coat. Splash in a little pasta water as needed to loosen and make it glossy. Top with sliced chicken and parsley. Serve immediately.
