Boil the pasta. Bring a large pot of salted water to a boil. Cook the pasta until just shy of al dente. Reserve 1/2 cup of pasta water, then drain and set aside.
Season the chicken. Pat the chicken dry.
Toss with 1–1.5 tbsp Cajun seasoning, a pinch of salt, and black pepper. Coat evenly so every piece gets flavor.
Brown the chicken. Heat olive oil in a large skillet over medium-high. Add the chicken in a single layer and cook 3–4 minutes per side until browned and cooked through.
Transfer to a plate and tent loosely with foil.
Sauté the veggies. Reduce heat to medium. Add butter, then the peppers and onion with a pinch of salt. Cook 4–5 minutes until softened and lightly charred at the edges.
Stir in garlic and cook 30 seconds until fragrant.
Deglaze and season. Pour in the chicken broth. Scrape up the browned bits. Stir in remaining Cajun seasoning, smoked paprika, and cayenne if using.
Simmer 2 minutes to reduce slightly.
Make it creamy. Lower the heat. Add the cream and bring to a gentle simmer, not a boil. Stir in Parmesan until melted and smooth.
Taste and adjust salt and pepper.
Combine pasta and chicken. Add the cooked pasta and chicken to the skillet. Toss to coat. If the sauce is thick, splash in reserved pasta water a little at a time until silky and clinging to the noodles.
Brighten and serve. Stir in lemon juice for a clean finish.
Garnish with parsley or scallions and extra Parmesan. Serve hot.