Cook spaghetti in salted water according to package instructions until al dente. Drain and set aside.
Season chicken pieces with Cajun seasoning and smoked paprika.
Heat olive oil in a large skillet over medium-high heat. Add chicken and cook until browned and cooked through, about 6 to 8 minutes. Remove chicken and set aside.
In the same skillet, melt butter over medium heat. Add minced garlic and sauté for 1 minute until fragrant.
Sprinkle flour into the skillet and whisk continuously for 1 to 2 minutes to form a roux.
Gradually whisk in milk and chicken broth, stirring constantly to prevent lumps. Bring the mixture to a gentle simmer.
Add cream cheese and 1 cup of mozzarella cheese, stirring until melted and smooth.
Stir in Parmesan cheese, then season the sauce with salt, black pepper, and red pepper flakes if using.
Return the cooked chicken to the sauce and simmer for 2 to 3 minutes to heat through.
Add the cooked spaghetti to the skillet and toss to coat in the sauce.
Sprinkle the remaining 1 cup of mozzarella cheese over the pasta, cover, and let sit for 2 to 3 minutes until the cheese melts.
Serve hot, optionally garnished with fresh parsley or extra Parmesan.