Pound chicken to about ½ inch thickness. Liberally season on both sides with Cajun seasoning. For spicier chicken, use more seasoning.
In a large pot, cook fettuccine pasta as directed on package. Drain and set aside.
* See Note for alternative if you don't have a steamer. In a large pot, add an inch of water and bring to a rolling boil. Place a steamer basket inside the pot and add the broccoli. Cover, turn the heat down to a medium-low, and cook for 6-10 minutes, until the broccoli is tender.
Melt 2 Tbsp butter in a cast iron skillet over medium-high heat. Add chicken and cook for 2-3 minutes per side. Chicken should be dark brownish black on the outside and cooked through, but still moist. The skillet may smoke during the blackening process. Don't be alarmed!
In a large, deep skillet melt 6 Tbsp of butter over medium heat. Add in minced garlic and sauté for about a minute until golden and fragrant. Slowly add in the cream and stir until well combined. Stir occasionally and bring to a bubbling boil, about 5 minutes. Add salt and pepper. Add in Parmesan cheese, ½ cup at a time, stirring between additions. Reduce heat.
In a small bowl, add ¼ cup of cream sauce and cornstarch. Mix until smooth and without lumps. Add cornstarch mixture into sauce and stir until well combined.
Mix noodles into sauce. Plate with broccoli and chicken on top.