Season the chicken. Pat chicken dry and toss with 1 tablespoon Cajun seasoning, a pinch of salt, and black pepper. This helps it brown well and taste bold from the start.
Cook the pasta. Bring a large pot of salted water to a boil.
Cook spaghetti until just shy of al dente (about 1 minute less than package). Reserve 1 cup of pasta water, then drain.
Sear the chicken. In a large skillet, heat olive oil over medium-high. Add chicken in a single layer and cook 5–7 minutes, stirring once or twice, until browned and cooked through.
Transfer to a plate.
Sauté the veggies. Lower heat to medium. Add butter, onion, and bell peppers. Cook 5–6 minutes until softened and lightly caramelized.
Stir in garlic and cook 30 seconds until fragrant.
Build the sauce base. Stir in tomato paste and cook 1 minute to deepen flavor. Add crushed tomatoes, chicken broth, smoked paprika, remaining Cajun seasoning, and red pepper flakes if using. Simmer 5 minutes to thicken slightly.
Add cream and cheese. Reduce heat to medium-low.
Stir in heavy cream and Parmesan until smooth. Taste and adjust salt, pepper, and Cajun seasoning. The sauce should be creamy, a little spicy, and well-seasoned.
Combine pasta and chicken. Return chicken and any juices to the skillet.
Add spaghetti and toss to coat. If the sauce seems thick, splash in reserved pasta water a little at a time until glossy and silky.
Finish and serve. Squeeze in a bit of lemon for brightness, then top with chopped parsley or green onions. Serve hot with extra Parmesan on the side.