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 saucepan, melt the butter over medium heat.
Whisk in the flour and cook for 1-2 minutes until it forms a smooth paste and starts to bubble, stirring constantly.
Gradually whisk in the milk and chicken broth, continuing to whisk until the mixture thickens and comes to a simmer, about 5-7 minutes.
Remove the sauce from heat and stir in the garlic powder, onion powder, salt, black pepper, 1 1/2 cups of the cheddar cheese, and 1/2 cup of the mozzarella cheese until melted and smooth.
In a large mixing bowl, combine the cooked noodles, shredded chicken, steamed broccoli, and cheese sauce. Stir gently until everything is evenly coated.
Transfer the mixture to a greased 9×13-inch baking dish.
Sprinkle the remaining 1/2 cup cheddar cheese, 1/2 cup mozzarella cheese, and grated Parmesan cheese evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese on top is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.