In a large pot, add water and a pinch of salt and bring to a boil. Add the fettuccine pasta and cook according to the package instructions until al dente, adding the broccoli florets during the last 2 minutes of cooking. Drain the pasta and broccoli. Reserve ½ cup of the pasta cooking water and set aside.
In a large skillet, melt 1 tablespoon of butter over medium-high heat. Add the chicken breast pieces and season with salt and pepper. Cook the chicken 2-3 minutes per side, until the outside has a golden crust. Remove the chicken from the skillet and set aside.
In the same skillet, melt the remaining 2 tablespoons of butter over medium heat. Add the minced garlic and cook until fragrant, about 1 minute. Pour in the heavy cream and simmer on low for 2-3 minutes. Slowly whisk in the grated Parmesan cheese until the sauce is smooth and creamy. Season with salt and pepper to your desired amount.
Add the cooked fettuccine and broccoli to the skillet with the Alfredo sauce, tossing to coat evenly. If the sauce is too thick, gradually add some of the reserved pasta cooking water until desired consistency is reached. Stir in the cooked chicken pieces.
Transfer the Chicken and Broccoli Fettuccine Alfredo to serving plates or a large platter. Garnish with chopped fresh parsley, red pepper flakes and some extra Parmesan cheese for a pop of color and flavor.