Slice each chicken breast horizontally right through the centre to make four even-sized breasts. In a large, shallow dish (or ziplock bag), whisk together the lime juice, oil and 1 tablespoon of the Cajun seasoning with the salt and black pepper. Add the chicken and toss to coat, then cover with cling film and marinate in the fridge for as long as you have time for (2 hours if you can, up to overnight). If you’re pushed for time, just rest the chicken at room temperature while you prepare the other ingredients.
If you marinated the chicken in the fridge, bring it to close to room temperature, then fry the breasts in a deep frying pan over a medium-high heat for a few minutes each side, or until lightly charred and just cooked through. Remove from the pan and set aside to rest, then slice into thin strips just before adding to the pasta.
Fry the mushrooms in the leftover marinade until they begin to soften (add a splash more oil if you need it). Add the onion and peppers and continue frying until golden (again, add a splash of oil if you need it). Add the garlic and fry for a minute or so longer. Add 1 tbsp Cajun Seasoning, fry for a minute, then melt in the butter. Pour in the chicken stock and cream, then bring to a gentle simmer. Allow to simmer until it begins to thicken (about 10 minutes), stirring occasionally.
While the sauce is simmering, cook the pasta in a saucepan of salted boiling water until al dente. Retain a cup of the starchy pasta water before draining.
Add the Parmesan and parsley to the sauce, then season to taste. Stir through the cooked pasta until coated in the sauce, then stir through the sliced chicken, along with any resting juices. Continue tossing until the sauce clings to the pasta, thinning it out with more pasta water if needed.
Serve with extra Parmesan and parsley and a squeeze of lime juice if desired.