Season the chicken breasts with Cajun seasoning and cook in olive oil over medium heat for 6-7 minutes on each side until cooked through. Let rest and slice.
Boil salted water, cook fettuccine according to package instructions, adding broccoli in the last 3 minutes. Drain and reserve ½ cup of pasta water.
In the same skillet, sauté minced garlic for 1 minute, then add heavy cream and bring to a simmer. Stir in Parmesan cheese until melted.
Combine cooked fettuccine and broccoli with the Alfredo sauce, tossing gently. Season with salt and pepper to taste.
Plate the dish, top with sliced chicken, and garnish with chopped parsley.