Preheat oven to 375°F (190°C). Grease a 9×13-inch baking dish.
Cook spaghetti in salted water until al dente. Drain and set aside.
In a large skillet, melt butter over medium heat. Sauté onion until soft, then add garlic and cook 30 seconds.
Stir in cream cheese, cream, and chicken broth until smooth.
Add cheddar, mozzarella, green chilies, paprika, cayenne, salt, and pepper. Stir until melted.
Fold in cooked chicken and spaghetti until evenly coated.
Transfer to baking dish, top with extra cheese, and bake 25–30 minutes until bubbly and golden.
Rest 5 minutes before serving.