Prep the chicken. Pat the chicken breasts dry and season both sides with salt and pepper. For quicker, more even cooking, you can pound them to an even thickness or slice them horizontally into cutlets.
Sear the chicken. Heat the olive oil in a large skillet over medium-high heat.
Add the chicken and cook 4–6 minutes per side, until golden and cooked through (165°F/74°C). Transfer to a plate and tent loosely with foil to rest.
Boil the pasta. Bring a large pot of salted water to a rolling boil. Add the fettuccine and cook until just shy of al dente according to package directions.
Reserve 1 cup of the starchy pasta water, then drain.
Build the base of the sauce. In the same skillet used for the chicken, lower the heat to medium and add the butter. Once melted and foamy, add the garlic. Cook 30–45 seconds, just until fragrant.
Do not brown the garlic.
Add the cream. Pour in the heavy cream and whisk to combine with the butter and garlic. Bring to a gentle simmer and cook 2–3 minutes to slightly thicken. Season with a pinch of salt, black pepper, and a tiny pinch of nutmeg if using.
Emulsify with cheese. Reduce the heat to low.
Gradually whisk in the Parmesan by small handfuls, letting each addition melt before adding more. You want a glossy, smooth sauce. If it gets too thick, loosen with a splash of reserved pasta water.
Toss the pasta. Add the drained fettuccine directly to the sauce.
Toss and turn until every strand is coated. Add more pasta water as needed to keep it silky and fluid, not gloopy.
Slice and add chicken. Slice the rested chicken across the grain. Fold it into the pasta or lay it on top for serving.
If you like, finish with a squeeze of lemon to brighten the richness.
Garnish and serve. Top with extra Parmesan, a sprinkle of parsley, and a few grinds of black pepper. Serve immediately while hot and creamy.