Boil the pasta: Bring a large pot of salted water to a boil.
Cook fettuccine until just shy of al dente (about 1 minute less than package time). Reserve 1 cup pasta water, then drain.
Blanch the broccoli: In the last 2 minutes of pasta cooking, add broccoli to the pot. Drain with the pasta. This keeps the broccoli bright and saves a pan.
Season the chicken: Pat chicken dry.
Season with 1/2 tsp salt and 1/2 tsp black pepper. If you like heat, add a pinch of red pepper flakes.
Sear the chicken: Heat olive oil in a large skillet over medium-high. Add chicken in a single layer and cook 3–4 minutes per side until golden and cooked through.
Transfer to a plate and tent with foil.
Build the sauce base: Reduce heat to medium. Add butter to the skillet. When melted, add garlic and cook 30–60 seconds until fragrant.
Don’t brown it.
Add cream: Pour in heavy cream. Stir and simmer gently 3–4 minutes until slightly thickened. Keep the heat moderate so it doesn’t scorch.
Finish with cheese: Lower the heat.
Stir in Parmesan a handful at a time until smooth. If it gets too thick, whisk in a splash of reserved pasta water. Taste and adjust salt and pepper.
Toss everything: Add the cooked pasta, broccoli, and chicken to the skillet.
Toss to coat. Use pasta water as needed to create a silky, scoopable sauce that clings to the noodles.
Brighten and serve: Stir in lemon juice if using. Top with more Parmesan and parsley.
Serve hot.