Boil the pasta: Bring a large pot of salted water to a rolling boil. Cook the pasta until just shy of al dente according to package directions. Reserve 1 cup of pasta water, then drain.
Blanch the broccoli: In the last 2 minutes of pasta cooking, add the broccoli florets to the pot. They should turn bright green and tender-crisp.
Drain with the pasta and set aside.
Season the chicken: Pat the chicken dry. Season with 1 teaspoon salt and 1/2 teaspoon black pepper. If you like heat, add a pinch of red pepper flakes.
Sear the chicken: Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high.
Add half the chicken in a single layer. Cook 2–3 minutes per side until golden and cooked through. Transfer to a plate.
Repeat with remaining oil, butter, and chicken.
Start the sauce: Reduce heat to medium. Add the remaining 1 tablespoon butter to the skillet. Stir in garlic and cook 30–45 seconds until fragrant, not browned.
Add cream and milk: Pour in the cream and milk.
Stir, scraping up any browned bits from the pan. Simmer gently for 2–3 minutes to slightly thicken. Do not boil hard.
Melt in Parmesan: Lower the heat.
Add Parmesan in small handfuls, stirring until smooth. Season with a pinch of nutmeg, salt, and plenty of black pepper. If the sauce seems too thick, stir in a splash of pasta water.
Toss it all together: Add the cooked pasta, broccoli, and chicken to the skillet.
Toss to coat, adding more pasta water a little at a time until the sauce is glossy and clings to the pasta.
Finish and serve: Taste and adjust seasoning. Add a sprinkle of lemon zest if using, then garnish with parsley and extra Parmesan. Serve right away.