Preheat the oven to 350°F (175°C).
Cook the egg noodles according to package instructions until al dente. Drain and set aside.
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large skillet, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté until the onion is translucent, about 3-4 minutes.
In a large mixing bowl, combine the cooked chicken, steamed broccoli, cooked noodles, sautéed onion and garlic, cream of mushroom soup, sour cream, mayonnaise, milk, salt, black pepper, and paprika. Mix well until all ingredients are evenly incorporated.
Stir in the shredded cheddar cheese and mozzarella cheese.
Transfer the mixture to a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the grated Parmesan cheese evenly over the top, then add the crushed buttery crackers as a topping.
Bake uncovered in the preheated oven for 30-35 minutes or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.