Cook linguine until al dente. Reserve pasta water before draining.
Season chicken with 1 tbsp Cajun seasoning, paprika, garlic powder, oregano, salt, and pepper.
Heat oil and 2 tbsp butter in a skillet. Sear chicken until golden-brown with crispy edges. Remove.
Melt remaining butter, add garlic and remaining Cajun seasoning, cook until fragrant.
Pour in heavy cream, whisk in cream cheese until smooth and pale orange.
Stir in 1 cup Parmesan and ½ cup mozzarella off the boil until thick and glossy. Loosen with pasta water.
Toss linguine through the sauce until fully coated.
Top with reserved mozzarella, cover 2 minutes until melted and stretchy.
Plate chicken beside the pasta, sprinkle remaining Parmesan, red pepper flakes, and parsley.