01 - Bring a large pot of salted water to a boil. Add fettuccine and cook according to package directions. Add broccoli florets during the last 3 minutes of cooking. Drain pasta and broccoli, reserving 1/2 cup of pasta water.
02 - Heat 1 tablespoon butter in a large skillet over medium-high heat. Season chicken with salt and pepper, then sauté until golden and cooked through, about 5 to 6 minutes. Remove chicken from skillet and set aside.
03 - Reduce heat to medium and add remaining butter to the skillet. Sauté garlic until fragrant, approximately 1 minute.
04 - Pour in heavy cream and milk, stirring to combine. Simmer gently for 2 to 3 minutes until slightly thickened.
05 - Stir in Parmesan cheese and nutmeg if using. Continue cooking and stirring until cheese melts and sauce is smooth. Adjust sauce consistency with reserved pasta water as needed.
06 - Return cooked fettuccine, broccoli, and chicken to the skillet. Toss gently to evenly coat with sauce. Adjust seasoning with salt and pepper to taste.
07 - Serve immediately, garnished with chopped parsley and additional Parmesan cheese if desired.