Bring a large pot of generously salted water to a boil. Cook the spaghetti according to package directions until al dente. Reserve 1/2 cup of the pasta water, then drain the spaghetti and toss it with 1 tablespoon olive oil to prevent sticking.
Season the chicken strips with Cajun seasoning, garlic powder, onion powder, smoked paprika, salt, and black pepper, tossing until evenly coated.
Heat the remaining 1 tablespoon olive oil and the butter in a large deep skillet over medium-high heat. Add the chicken in an even layer and cook for 6 to 8 minutes, stirring occasionally, until browned and cooked through to an internal temperature of 165°F. Transfer the chicken to a plate.
Reduce the heat to medium. Add the minced garlic to the same skillet and cook for about 30 seconds, stirring constantly, until fragrant.
Pour in the chicken broth and scrape up the browned bits from the bottom of the skillet. Stir in the heavy cream and bring the mixture to a gentle simmer.
Add the cream cheese and Velveeta, stirring until fully melted and smooth. Stir in the Parmesan, mozzarella, and Italian seasoning. Cook for 2 to 3 minutes, stirring often, until the sauce is creamy and slightly thickened.
Return the chicken and any collected juices to the skillet. Add the cooked spaghetti and toss until the pasta and chicken are thoroughly coated. Add a splash of reserved pasta water if needed to loosen the sauce. Taste and add more Cajun seasoning if desired.
Remove from the heat and sprinkle with parsley and additional Parmesan. Serve hot.