Preheat the oven to 350°F (175°C).
Steam the broccoli florets until just tender, about 4-5 minutes, then drain and set aside.
In a medium saucepan, melt the butter over medium heat.
Add the chopped onion and cook for 3-4 minutes until softened.
Whisk in the all-purpose flour and cook for 1-2 minutes, stirring constantly to form a roux.
Gradually whisk in the whole milk, continuing to stir until the sauce thickens, about 5 minutes.
Remove the sauce from heat and stir in garlic powder, onion powder, salt, and black pepper.
In a large mixing bowl, combine the shredded cooked chicken, steamed broccoli, cheddar cheese, mozzarella cheese, and the prepared creamy sauce. Mix well.
Transfer the mixture into a greased 9×13-inch baking dish.
Sprinkle the plain breadcrumbs evenly over the top of the casserole.
Bake in the preheated oven for 30 minutes or until the casserole is bubbly and the breadcrumbs are golden brown.
Remove from oven and let it rest for 5 minutes before serving.