Preheat the oven to 375°F (190°C).
Steam the broccoli florets in a steamer basket over boiling water for 4-5 minutes until tender-crisp. Drain and set aside.
In a medium saucepan, melt the butter over medium heat.
Add the chopped onion to the melted butter and sauté for 3-4 minutes until translucent.
Whisk in the 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 and begins to bubble, about 5-7 minutes.
Remove the sauce from heat and stir in the cheddar cheese, mozzarella cheese, garlic powder, onion powder, salt, and black pepper until the cheese is melted and the sauce is smooth.
In a large mixing bowl, combine the cooked chicken, steamed broccoli, and cheese sauce. Mix gently to combine.
Transfer the mixture to a greased 9×13-inch baking dish and spread evenly.
In a small bowl, mix the breadcrumbs, grated Parmesan cheese, and olive oil until the crumbs are evenly coated.
Sprinkle the breadcrumb mixture evenly over the casserole.
Bake uncovered in the preheated oven for 25-30 minutes, or until the top is golden brown and the casserole is bubbly.
Remove from the oven and let it rest for 5 minutes before serving.