01 - Bring a large pot of salted water to a boil. Add fettuccine and cook according to package instructions. Add broccoli florets during the final 3 minutes of cooking. Drain pasta and broccoli, reserving 1/2 cup of pasta cooking water.
02 - While pasta cooks, heat 1 tablespoon 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 skillet heat to medium and add remaining butter. Sauté minced garlic until fragrant, approximately 1 minute.
04 - Pour in heavy cream and whole milk, stirring to combine. Allow to simmer gently for 2 to 3 minutes until sauce slightly thickens.
05 - Stir in grated Parmesan cheese and nutmeg if desired. Continue to cook, stirring, until cheese melts and sauce reaches a smooth consistency. Thin sauce by gradually adding reserved pasta water if necessary.
06 - Return cooked fettuccine, broccoli, and chicken to the skillet. Toss gently to coat evenly with sauce. Adjust seasoning with salt and pepper to taste.
07 - Serve immediately, garnished with chopped fresh parsley and additional Parmesan cheese if preferred.