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 until lightly golden, stirring constantly.
Gradually whisk in the milk, ensuring no lumps remain.
Cook the sauce, stirring frequently, until it thickens and coats the back of a spoon, 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 cheddar cheese and all of the mozzarella 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 gently to combine.
Transfer the mixture to a greased 9×13-inch baking dish.
Sprinkle the remaining 1/2 cup cheddar cheese evenly over the top.
Sprinkle the grated Parmesan cheese and breadcrumbs evenly over the casserole.
Bake uncovered in the preheated oven for 25-30 minutes, until the top is golden brown and bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.