Bring a large pot of salted water to a boil and cook 12 ounces pasta according to package directions until al dente. Reserve ½ cup pasta water, then drain.
Season 1½ pounds chicken with 2 teaspoons Cajun seasoning, ½ teaspoon salt, and ½ teaspoon black pepper.
Heat a large skillet over medium-high heat and add 1 tablespoon olive oil and 1 tablespoon butter. Add the chicken and cook for 5 to 6 minutes, stirring occasionally, until golden and cooked through. Remove the chicken from the pan and set aside.
Reduce the heat to medium. Add 3 tablespoons butter to the same skillet. Once melted, stir in 5 minced garlic cloves and cook for 30 seconds, just until fragrant.
Add 1 tablespoon Cajun seasoning, 1 teaspoon smoked paprika, and ½ teaspoon chili flakes if using. Stir for 20 seconds to bloom the spices.
Pour in 2 cups heavy cream, stirring constantly. Bring the sauce to a gentle simmer and cook for 4 to 5 minutes, until slightly thickened.
Stir in 1 cup grated Parmesan cheese a little at a time, mixing until the sauce is smooth and glossy. If the sauce is too thick, add reserved pasta water a splash at a time.
Return the cooked chicken to the pan along with the drained pasta and any optional vegetables. Toss everything together until fully coated in sauce. Season with ½ teaspoon salt and ½ teaspoon black pepper, adjusting to taste. Let everything cook together for 2 to 3 minutes, until hot and ultra creamy.
Remove from heat and garnish with fresh parsley and extra Parmesan cheese before serving.