Horizontally slice the chicken breasts right through the centre to make 4 even-sized breasts. In a large shallow dish (or zip-lock bag) whisk together 1 tbsp oil, lime juice & cajun seasoning with 1/4 tsp salt & 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 (up to overnight). If you're pushed for time just rest at room temp as you prep the other ingredients.
Bring the chicken to close room temp if you marinated it in the fridge, then add to a large pan over medium-high heat. Fry for a few mins each side, or until lightly charred and just through the centre. Rest to one side until needed, then slice into thin strips just before adding to pasta.
Fry mushrooms in the leftover oil until they begin to soften (top up with a dash more oil if you need to). Add peppers & onion and continue frying until golden (again, add a dash of oil if you need it). Add garlic and fry for a minute or so longer. Add 1 tbsp Cajun Seasoning, fry for a minute, then melt in 2 tbsp butter.
Pour in the chicken stock and cream, then bring to a gentle simmer. Allow to simmer until it begins to thicken, stirring occasionally (approx 10mins). At this point pop the pasta in salted boiling water and cook until al dente. Save a cup of starchy pasta water before draining.
To the sauce add parmesan, parsley and seasoning to taste. Stir through the cooked pasta until coated in the sauce, then stir through the sliced chicken, alongside any resting juices. Continue tossing until the sauce wraps around and clings to the pasta, thinning out with more pasta water if needed. Serve with extra parmesan & parsley and a squeeze of lime juice if desired.