01 - Bring a large pot of salted water to a boil. Add fettuccine and cook according to package instructions. Add broccoli florets during the last 3 minutes of pasta cooking. Drain, reserving 1/2 cup pasta water.
02 - While the pasta cooks, heat 1 tablespoon butter in a large skillet over medium-high heat. Add chicken pieces, season with salt and pepper, and sauté until golden and cooked through, about 5–6 minutes. Remove chicken and set aside.
03 - In the same skillet, lower heat to medium and add remaining butter. Sauté garlic until fragrant, about 1 minute.
04 - Pour in heavy cream and milk, stirring to combine. Simmer for 2–3 minutes until slightly thickened.
05 - Stir in Parmesan cheese and nutmeg (if using). Cook, stirring, until cheese is melted and sauce is smooth. If sauce is too thick, add reserved pasta water a little at a time.
06 - Add cooked fettuccine, broccoli, and chicken to the skillet. Toss gently to coat pasta in sauce. Adjust seasoning with salt and pepper if needed.
07 - Serve immediately, garnished with chopped parsley and extra Parmesan if desired.