Boil the pasta. Bring a large pot of well-salted water to a boil. Add fettuccine and cook until just al dente, 1 minute shy of package time. Reserve 1 cup pasta water, then drain.

Prep the chicken. Slice chicken into thin strips. Pat dry and season with 1 teaspoon salt and 1/2 teaspoon black pepper.

Sear the chicken. Heat a large skillet over medium-high. Add olive oil and 1 tablespoon butter. Cook chicken in a single layer until golden and just cooked through, about 4–6 minutes, stirring once or twice. Transfer to a plate.

Sauté the garlic. Lower heat to medium. Add remaining 2 tablespoons butter to the same skillet. Stir in garlic and cook until fragrant, about 30 seconds. Don’t brown it.

Build the sauce. Pour in chicken broth and simmer for 1 minute, scraping up browned bits. Stir in heavy cream and a pinch of red pepper flakes if using. Simmer gently for 2–3 minutes to thicken slightly.

Add the cheese. Reduce heat to low. Sprinkle in Parmesan a handful at a time, whisking until smooth. If it seems too thick, add a splash of reserved pasta water. Taste and season with salt and pepper.

Combine. Return chicken (and juices) to the skillet. Toss in drained pasta. Add a little pasta water as needed to create a silky, glossy sauce that coats the noodles.

Finish and serve. Add a small squeeze of lemon if you like. Garnish with parsley. Serve hot with extra Parmesan at the table.