Pat the chicken dry and flatten slightly so it cooks evenly. Rub chicken with 1 tbsp olive oil and coat fully in Cajun seasoning until bronzed all over.
Bring a big pot of salted water to a boil. Cook pasta until al dente (box time minus 1 minute). Reserve a cup of pasta water, then drain and drizzle with olive oil to avoid clumping.
In a large pan over medium-high heat, add 2 tbsp neutral oil and 1 tbsp butter. Sear chicken 4 minutes per side until golden. Lower heat, add 2 tbsp butter, and baste until cooked through. Remove and set aside.
In the same pan, melt 2 tbsp of butter, add diced onion and sauté until softened and golden (about 5 minutes). Toss in bell peppers and extra seasonings, sauté 6-7 minutes until soft. Add minced garlic, cook 2 minutes.
Lower to medium-low. Stir in cream and let it gently simmer. Add spinach and parmesan, mix for a minute. Add pasta and ¼ cup pasta water, toss well over low heat for 1-2 minutes to emulsify. If sauce is too thick, splash in more pasta water.
Slice chicken, lay on top, and serve with fresh basil and additional parmesan.
