Boil pasta according to package directions. Be sure to reserve 1 cup of pasta water before straining.
While the pasta is cooking, pat the chicken breasts dry with a paper towel. Divide 1 tbsp of Cajun seasoning and 1/2 tsp salt on both sides of the chicken, pressing the seasoning in with your fingers.
Heat the olive oil in a medium pan over medium heat. Add the chicken breasts and cook undisturbed for 3-5 minutes per side, depending on the thickness of your chicken breasts. Using a food thermometer, check to make sure the internal temperature is 165 ℉. Set chicken breasts aside on a clean plate to cool while you make the sauce.
In the same pan you used to cook the chicken, melt the butter over medium-high heat. Add in the onion, celery, and bell pepper. Cook for 3-5 minutes or until veggies are very soft. Add in the garlic and cook for 1 additional minute.
Add in the reserved pasta water (or chicken broth) and let simmer for 1-2 minutes, scraping the bottom of the pan to deglaze if necessary.
Reduce to low heat. Add in the heavy cream, parmesan cheese, and remaining Cajun seasoning with salt. Let it cook just until the cheese is melted and the sauce is thick. Add in the cooked pasta and toss to coat.
Slice the chicken breasts. Add the chicken with their juices to the pot and toss again. Garnish with parsley, serve warm, and enjoy!