Bring a large pot of well-salted water to a boil for the pasta. While it heats, pat the chicken dry and season both sides with salt and pepper.
Heat olive oil in a large skillet over medium-high heat. Add the chicken and cook for 6–7 minutes per side, until golden and cooked through (165°F/74°C). Transfer to a plate and let rest 5 minutes, then slice into strips.
Cook the fettuccine until al dente according to the package directions. Reserve about 1/2 cup pasta water, then drain (do not rinse).
Reduce the skillet heat to medium. Add butter and, once melted, sauté the minced garlic for about 1 minute until fragrant (don’t brown it).
Pour in the heavy cream and bring to a gentle simmer. Add the Parmesan in handfuls, stirring constantly until melted and the sauce thickens. Season with 1/4 tsp salt and 1/4 tsp pepper (and a pinch of nutmeg if using).
Add the drained fettuccine to the sauce and toss until evenly coated. If needed, loosen with a splash of reserved pasta water.
Serve topped with sliced chicken, extra Parmesan, parsley, and freshly cracked black pepper.