Bring a large pot of salted water to a rolling boil. Add the spaghetti and cook according to package directions until al dente. Drain thoroughly, drizzle lightly with olive oil to prevent sticking, and set aside.

In a large skillet, heat the olive oil and 2 tablespoons of butter over medium-high heat. Add the bite-sized chicken pieces along with Cajun seasoning, paprika, salt, and black pepper. Cook for 6–8 minutes, stirring occasionally, until the chicken is golden brown and fully cooked through. Transfer the cooked chicken to a plate and set aside.

In the same skillet, melt the remaining 2 tablespoons of butter over medium heat. Stir in the minced garlic and sauté for approximately 1 minute until fragrant. Pour in the heavy cream and chicken broth. Bring the mixture to a gentle simmer, stirring continuously.

In a small bowl, whisk together the cornstarch and cold water until a smooth slurry forms. Slowly stir this slurry into the simmering cream mixture, whisking constantly until the sauce visibly thickens, which should take about 2–3 minutes.

Reduce the heat to low. Stir in the shredded mozzarella and grated Parmesan cheeses until they are completely melted and the sauce becomes creamy and smooth. Add the cooked spaghetti and seared chicken into the skillet with the sauce. Use tongs to toss everything together, ensuring the pasta is fully coated and the chicken is well-integrated into the creamy sauce.

Allow the dish to simmer gently for an additional 2–3 minutes to allow the flavors to meld. Serve immediately, artfully swirling the spaghetti onto individual plates and topping generously with the luscious sauce and juicy chicken.