Bring a large pot of salted water to a boil. Cook the fettuccine according to the package directions until al dente. Reserve 1/3 cup of the pasta water, then drain the pasta and set it aside.
In a medium mixing bowl, combine 2 teaspoons Cajun seasoning with 1 tablespoon of olive oil. Rub evenly all over the chicken breasts. Let the chicken marinate while you prep the vegetables.
Heat 1 tablespoon of olive oil in a large non-stick skillet over medium-high heat. Once very hot, add the chicken and cook, undisturbed, for 3-5 minutes, until nicely browned on one side. Flip the chicken, cover the skillet, and reduce the heat to medium. Cook for 5-7 minutes, depending on the thickness of the chicken, until it's cooked through. Transfer to a cutting board and let rest for at least 5 minutes before slicing.
Add 1 tablespoon of olive oil to the remaining oil in the skillet, then turn the heat to medium-high. Add the bell peppers, onions, and ½ teaspoon of Cajun seasoning and saute for 1 minute. Transfer to a plate.
Wipe out the skillet, then pour in the sun-dried tomato oil and place it over medium heat. Add the sun-dried tomatoes and garlic to the oil and saute for 1 minute. Sprinkle in the flour and cook for 2 minutes, stirring constantly.
Turn the heat to low and stir in chicken broth. Whisk cornstarch and milk together to make a slurry. Pour the slurry into the skillet along with tomato paste, ½ teaspoon of Cajun seasoning, salt, and pepper. Bring the sauce to a simmer. Continue to cook and stir until the sauce is thickened (but not overly thick), about 5 minutes. Stir in the cream cheese until melted. Add the Parmesan cheese and whisk until melted, and finally whisk in the mozzarella until melted.
Add the cooked pasta and peppers to the sauce. Toss until evenly coated, adding splashes of reserved pasta water a little at a time until your desired sauce consistency is reached. Top the pasta with sliced chicken or toss the chicken in with the pasta.
Garnish individual servings with plenty of freshly squeezed lime juice, salt, and pepper. Sprinkle with fresh parsley and a little shredded parmesan if desired. Serve warm.
Notes
You will need sun-dried tomatoes packed in oil for this recipe. The infused oil will be used for the sauce, so skip the dry-packed tomatoes this time.