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. Drain, reserving 1/2 cup pasta water.
02 - Heat 1 tablespoon butter over medium-high heat in a large skillet. Add chicken pieces, season with salt and pepper, and cook until golden and cooked through, about 5 to 6 minutes. Remove chicken and set aside.
03 - Reduce heat to medium, add remaining butter to the same skillet, and sauté minced garlic until fragrant, approximately 1 minute.
04 - Pour in heavy cream and whole milk, stirring to combine. Simmer for 2 to 3 minutes until slightly thickened.
05 - Stir in freshly grated Parmesan cheese and nutmeg if using. Cook until cheese melts and sauce is smooth. Adjust thickness with reserved pasta water as needed.
06 - Add cooked fettuccine, broccoli, and chicken back into the skillet. Toss gently to coat evenly with sauce. Adjust seasoning with salt and pepper to taste.
07 - Serve immediately, garnished with chopped parsley and additional Parmesan cheese as desired.