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 saucepan, melt the butter over medium heat.
Whisk in the flour and cook for 1-2 minutes, stirring constantly, to form a roux.
Gradually whisk in the milk, continuing to whisk until the sauce thickens and is smooth, about 5-7 minutes.
Remove the sauce from heat and stir in garlic powder, onion powder, salt, and black pepper.
Add 1 1/2 cups of the cheddar cheese, all the mozzarella cheese, and 1/4 cup of the Parmesan cheese to the sauce. Stir until the cheese is melted and the sauce is smooth.
In a large mixing bowl, combine the cooked noodles, shredded chicken, steamed broccoli, and cheese sauce. Mix well to combine.
Transfer the mixture to a greased 9×13-inch baking dish and spread evenly.
In a small bowl, mix the breadcrumbs with the remaining 1/2 cup cheddar cheese and 1/4 cup Parmesan cheese.
Sprinkle the breadcrumb and cheese mixture evenly over the casserole.
Bake in the preheated oven for 25-30 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.