Bring a large pot of salted water to a rolling boil and cook fettuccine until tender but firm.
Add broccoli florets to the pasta water during the last 3 minutes of cooking.
Drain pasta and broccoli, reserving some pasta water.
Season cubed chicken evenly with salt and black pepper.
Heat olive oil in a skillet over medium-high heat and sauté chicken for 6 to 7 minutes until golden and cooked through.
Remove chicken from the skillet and set aside.
Lower heat and melt butter in the same skillet, then add garlic and cook briefly until fragrant.
Pour in heavy cream and simmer gently until slightly reduced, about a few minutes.
Stir in Parmesan cheese until the sauce is smooth and velvety.
Add pasta and broccoli to the skillet, tossing to coat with the sauce, adjusting consistency with reserved pasta water if needed.
Fold in the cooked chicken and heat through for another minute.
Serve immediately, garnished with fresh parsley and extra Parmesan cheese if desired.