In a large pot, bring salted water to a boil. Add the fettuccine or linguine and cook according to package instructions until al dente. Drain and set aside.
In a mixing bowl, toss the sliced chicken breasts with the Cajun seasoning, ensuring they are evenly coated.
In a large skillet, heat 2 tablespoons of olive oil over medium-high heat. Add the seasoned chicken and cook for about 5-7 minutes until the chicken is golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining tablespoon of olive oil, and then add the minced garlic and sliced bell pepper. Sauté for around 3-4 minutes until the vegetables soften.
Pour in the chicken broth and bring it to a simmer. Then, lower the heat and add the heavy cream. Stir well to combine and let it cook for 2-3 minutes until the sauce thickens slightly.
Add the cooked pasta and the sautéed chicken back into the skillet. Toss everything together until the pasta is well coated with the creamy Cajun sauce. Stir in the grated Parmesan cheese and season with salt and pepper to taste.
Remove from heat and let sit for a minute to thicken, if desired. Serve hot, garnished with freshly chopped parsley on top.