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 bright green, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the condensed cream of mushroom soup, sour cream, mayonnaise, milk, garlic powder, onion powder, salt, and black pepper. Mix until smooth.
Add the cooked chicken, steamed broccoli, and cooked noodles to the bowl. Stir gently to combine all ingredients evenly.
Fold in 1/2 cup of shredded cheddar cheese and 1/4 cup of grated Parmesan cheese into the mixture.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
In a small bowl, melt the unsalted butter and mix it with the crushed buttery crackers.
Sprinkle the cracker mixture evenly over the casserole.
Top the casserole with the remaining 1/2 cup shredded cheddar cheese and 1/4 cup grated Parmesan cheese.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from oven and let it rest for 5 minutes before serving.