Preheat the oven to 350°F (175°C).
Bring a large pot of salted water to a boil. Add the egg noodles and cook according to package instructions until al dente, about 7-8 minutes. Drain and set aside.
While the noodles cook, steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the sour cream, mayonnaise, condensed cream of chicken soup, milk, garlic powder, onion powder, salt, and black pepper. Stir until smooth.
Add the cooked noodles, shredded chicken, steamed broccoli, and 1 1/2 cups of shredded cheddar cheese and all the mozzarella cheese into the sauce mixture. Stir gently until everything is evenly coated.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
In a small bowl, mix the crushed buttery crackers with the melted butter until well combined. Sprinkle this mixture evenly over the casserole.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly on top.
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.