Cook fettuccine in salted boiling water until al dente. Reserve ½ cup pasta water, then drain.

Season chicken with salt and pepper. Melt 1 tbsp butter in a large skillet over medium-high heat. Sear chicken 5 to 6 minutes per side until golden and cooked through (165°F/74°C). Transfer to a cutting board, rest, then slice.

Lower heat to medium. Melt remaining 2 tbsp butter in the same skillet. Sauté garlic 30 seconds. Add heavy cream and simmer 2 to 3 minutes until slightly thickened. Off heat, whisk in Parmesan gradually. Stir in reserved pasta water to desired consistency.

Add cooked pasta and sliced chicken to the sauce. Toss to coat, adding more pasta water if needed. Serve immediately with extra Parmesan and parsley.