Boil the pasta. Bring a large pot of salted water to a boil. Cook the pasta until just al dente. Reserve 1/2 cup pasta water, then drain and set aside.

Season the chicken. Pat the chicken dry. Toss with 1–1.5 tablespoons Cajun seasoning, a pinch of salt, and a few cracks of black pepper.

Sear the chicken. Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Add the chicken in a single layer and cook until browned and cooked through, about 5–7 minutes. Transfer to a plate and tent with foil.

Sauté the vegetables. In the same skillet, add remaining olive oil and butter. Stir in the peppers and onion with a pinch of salt. Cook until tender and slightly charred at the edges, 5–6 minutes. Add the garlic and cook 30 seconds.

Bloom the spices. Sprinkle in 1 tablespoon Cajun seasoning and the tomato paste. Stir and cook 1 minute to deepen the flavor.

Build the sauce. Pour in the chicken broth and scrape up browned bits. Simmer 2 minutes. Lower the heat and stir in the cream. Let it gently bubble for 2–3 minutes until slightly thickened.

Add cheese and adjust. Stir in the Parmesan until melted. Taste and adjust seasoning with salt, pepper, or more Cajun seasoning. If you want extra heat, add a pinch of red pepper flakes.

Bring it together. Return the chicken and any juices to the pan. Add the drained pasta and toss to coat. If the sauce is too thick, loosen with reserved pasta water, a splash at a time, until glossy and silky.

Finish and serve. Squeeze in a little lemon juice for brightness. Top with chopped parsley or green onions. Serve hot with extra Parmesan.