Bring a large pot of salted water to a boil.
Season chicken with 1 tsp salt and the black pepper.
Heat olive oil and 1 tbsp butter in a large skillet over medium-high heat.
Sear chicken until golden and cooked through, 4–5 minutes per side. Transfer to rest.
Cook fettuccine until just shy of al dente. Reserve 1/2 cup pasta water and drain.
Lower skillet heat to medium. Melt remaining butter, then stir in garlic for 30 seconds.
Pour in cream and broth. Simmer gently, stirring, until slightly thickened, 3–4 minutes.
Whisk in Parmesan gradually until smooth. Add nutmeg and a pinch of the remaining salt.
Slice rested chicken thinly. Toss pasta into the sauce, loosening with pasta water as needed.
Fold in chicken. Finish with parsley and lemon juice. Taste and adjust seasoning.
Serve hot with extra Parmesan and a pinch of red pepper flakes if desired.