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 cooking. Drain pasta and broccoli, reserving 1/2 cup of the pasta cooking water.
02 - While pasta cooks, heat 1 tablespoon of butter in a large skillet over medium-high heat. Season chicken pieces 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, add remaining tablespoon of butter to the skillet, then sauté minced garlic until fragrant, about 1 minute.
04 - Pour in heavy cream and whole milk, stirring to combine. Allow to simmer gently for 2 to 3 minutes until slightly thickened.
05 - Stir in grated Parmesan cheese and ground nutmeg if using. Cook while stirring until cheese melts and sauce is smooth. If sauce is too thick, gradually add reserved pasta water until desired consistency is reached.
06 - Add cooked fettuccine, broccoli, and chicken back to the skillet. Toss gently to fully coat ingredients with the sauce. Adjust seasoning with salt and pepper to taste.
07 - Serve immediately, garnished with chopped fresh parsley and additional Parmesan cheese if desired.