Preheat the oven to 350°F. Lightly grease a 9x13-inch baking dish with nonstick cooking spray or butter.
Bring a large pot of salted water to a boil. Cook the spaghetti until it is just shy of al dente, following the package directions as a guide. Drain the pasta and set it aside. Do not overcook it because it will continue cooking in the oven.
Melt the butter in a large skillet or Dutch oven over medium heat. Add the diced onion and green bell pepper. Cook for 4–5 minutes, stirring occasionally, until the vegetables begin to soften.
Add the minced garlic and cook for about 30 seconds, just until fragrant. Do not allow the garlic to brown.
Stir in the cream of chicken soup, cream of mushroom soup, undrained Rotel, chicken broth, sour cream, smoked paprika, garlic powder, onion powder, black pepper, Cajun seasoning, and a small pinch of salt. Mix until the sauce is smooth and evenly combined.
Fold in the shredded chicken, 1 cup of the cheddar cheese, and the Monterey Jack cheese. Stir until the cheese begins melting into the sauce.
Add the drained spaghetti and toss until the pasta is completely coated. If the mixture is too thick, stir in another small splash of chicken broth.
Transfer the chicken spaghetti mixture to the prepared baking dish and spread it evenly. Sprinkle the remaining 1 cup of cheddar cheese over the top.
Bake uncovered for 20–25 minutes, until the casserole is hot and bubbly and the cheese has melted. Let it rest for 5–10 minutes before serving. Garnish with fresh parsley, if desired.