Bring a large pot of water to a boil. Add the pasta and cook according to package instructions. Reserve 2 cups of pasta water before draining. Set pasta aside.
Dice the onion and cut the broccoli into small florets.
Cut each chicken breast in half lengthwise and season with Cajun seasoning and pepper.
Heat oil in a large pan over medium-high heat. Add the chicken and cook until fully cooked through, about 5-7 minutes per side. Remove from the pan, shred using two forks, and set aside.
In the same pan, melt the margarine or butter. Stir in the flour and cook for 1 minute. Add the garlic and onion, cooking until softened and fragrant.
Pour in the cooking cream and bring to a gentle boil. Stir in the shredded cheese and season with pepper to taste. Slowly add half of the reserved pasta water, stirring until the sauce thickens.
Add the cooked chicken, broccoli, and pasta to the pan. Toss everything together, cover, and cook over medium heat until the broccoli is tender but still crisp (al dente), about 3-5 minutes.
Adjust seasoning if needed. Add more pasta water to reach your desired sauce consistency.
Serve warm and enjoy!