Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch baking dish.
Steam the broccoli florets until just tender, about 4-5 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Add the chopped onion and cook until softened, about 3 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the onion and garlic mixture. Stir constantly for 1-2 minutes to cook the flour.
Gradually whisk in the milk and chicken broth, continuing to whisk until the sauce thickens, about 5 minutes.
Remove the sauce from heat and stir in salt, black pepper, paprika, and dried thyme.
In a large mixing bowl, combine the shredded chicken, steamed broccoli, cheddar cheese, mozzarella cheese, and the prepared sauce. Mix until evenly coated.
Transfer the mixture to the prepared baking dish and spread evenly.
Sprinkle the breadcrumbs evenly over the top of the casserole.
Bake in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the breadcrumbs are golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.