Cook the linguine in salted water until al dente. Save about ⅓ cup of the pasta water, then drain and set the pasta aside.
Season the chicken with 1 tablespoon Cajun seasoning, smoked paprika, garlic powder, onion powder, ½ teaspoon salt, and black pepper.
Heat the olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Add the chicken and cook for 6 to 8 minutes, turning a few times, until browned and cooked through.
Add 1 tablespoon butter and half of the garlic to the skillet. Stir for about 30 seconds so the chicken gets coated in the garlic butter, then move the chicken to a plate and keep it separate.
Lower the heat to medium. Add the last tablespoon of butter and the remaining garlic to the same skillet. Cook briefly, just until the garlic smells fragrant.
Pour in the heavy cream and chicken broth. Stir in the remaining ½ tablespoon Cajun seasoning and ¼ teaspoon salt.
Add the Velveeta and stir until melted and smooth. Mix in the mozzarella and Parmesan and keep stirring until the sauce turns thick and creamy.
Add the drained linguine and toss until the noodles are fully coated. If the sauce feels too thick, add a little reserved pasta water.
Spoon the creamy linguine onto plates and arrange the Cajun chicken beside it. Finish with chopped parsley and serve while hot.