Preheat the oven to 375°F (190°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large skillet, heat the olive oil over medium heat. Add the diced onion, celery, and carrots. Sauté for 5-7 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Sprinkle the flour over the vegetables in the skillet and stir well to coat. Cook for 2 minutes, stirring constantly.
Gradually whisk in the almond milk and coconut milk, stirring continuously to avoid lumps.
Add the nutritional yeast, Dijon mustard, dried thyme, smoked paprika, onion powder, garlic powder, salt, and black pepper. Stir and cook until the sauce thickens, about 5 minutes.
In a large mixing bowl, combine the cooked chicken, steamed broccoli, cooked brown rice, and the prepared sauce. Mix well to combine all ingredients evenly.
Transfer the mixture into a greased 9×13 inch baking dish.
Sprinkle the gluten-free breadcrumbs evenly over the top of the casserole.
Bake in the preheated oven for 25-30 minutes, until the casserole is bubbly and the breadcrumbs are golden brown.
Remove from the oven and let it cool for 5 minutes before serving.