Bring a large pot of well-salted water to a boil. Cook the fettuccine until al dente according to the package directions. Reserve 1/2 cup pasta water, then drain.

Season the chicken with kosher salt and black pepper. Heat the olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Cook the chicken for 5–7 minutes per side, or until the thickest part reaches 165°F. Transfer to a cutting board and rest for 5 minutes.

Reduce the skillet heat to medium. Add the 3 tablespoons butter and garlic. Cook for about 30 seconds, until fragrant.

Add the heavy cream, nutmeg, and black pepper. Bring to a gentle simmer and cook for 3–4 minutes, until slightly reduced.

Reduce the heat to low. Gradually whisk in the Parmesan until smooth. Add a splash of reserved pasta water if needed.

Add the drained fettuccine and toss until coated. Add more pasta water a little at a time if the sauce becomes too thick.

Slice the chicken and arrange it over the fettuccine. Finish with parsley and extra Parmesan. Serve immediately.