Season chicken breasts on both sides with Cajun seasoning, salt, and pepper. Heat olive oil in a large skillet over medium-high heat. Sear chicken for 6–7 minutes per side until golden and cooked through. Remove from skillet, rest briefly, then slice into strips.
Bring a large pot of salted water to a boil. Cook pasta according to package directions until al dente. Reserve 1 cup of pasta water, then drain.
In the same skillet, melt butter over medium heat. Add garlic and cook until fragrant, about 30 seconds. Stir in Cajun seasoning and smoked paprika. Pour in heavy cream and chicken broth, simmering until slightly thickened. Stir in Parmesan cheese until smooth.
Add cooked pasta to the sauce, tossing to coat evenly. Use reserved pasta water to loosen sauce if needed. Gently fold in sliced chicken. Garnish with parsley and extra Parmesan if desired. Serve immediately.
