Preheat the oven to 350°F (175°C).
Rinse the jasmine rice under cold water until the water runs clear.
In a medium saucepan, bring 3 cups of water to a boil. Add the jasmine rice, reduce heat to low, cover, and simmer for 15 minutes, or until the rice is tender and water is absorbed.
While the rice cooks, steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes, stirring constantly to form a roux.
Gradually whisk in the milk, continuing to whisk until the sauce thickens, about 3-5 minutes.
Remove the sauce from heat and stir in garlic powder, onion powder, salt, and black pepper.
In a large mixing bowl, combine the cooked jasmine rice, shredded chicken, steamed broccoli, sour cream, cheddar cheese, and half of the mozzarella cheese. Pour the sauce over the mixture and stir until well combined.
Transfer the mixture to a greased 9×13 inch casserole dish. Sprinkle the remaining mozzarella cheese and Parmesan cheese evenly on top.
Bake uncovered in the preheated oven for 20-25 minutes, or until the casserole is bubbly and the cheese on top is golden brown.
Remove from oven and let it cool for 5 minutes before serving.