Preheat the oven to 350°F (175°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a medium skillet, heat olive oil and butter over medium heat. Add the chopped onion and minced garlic and sauté until softened and fragrant, about 3-4 minutes.
In a large mixing bowl, combine the condensed cream of mushroom soup, sour cream, milk, sautéed onion and garlic, salt, black pepper, and dried thyme. Mix well to create the creamy sauce.
In a 9×13-inch casserole dish, spread half of the cooked rice evenly on the bottom layer.
Top the rice layer with half of the shredded chicken evenly distributed.
Add a layer of half the steamed broccoli florets over the chicken.
Pour half of the creamy sauce evenly over the broccoli layer.
Sprinkle half of the shredded cheddar and mozzarella cheese over the sauce.
Repeat the layering process: spread the remaining rice, then the remaining chicken, followed by the remaining broccoli.
Pour the remaining creamy sauce over the top layer of broccoli.
Top the casserole with the remaining shredded cheddar and mozzarella cheese.
Cover the casserole dish with aluminum foil and bake in the preheated oven for 30 minutes.
Remove the foil and bake for an additional 15 minutes, or until the cheese is melted and bubbly and the casserole is heated through.
Let the casserole rest for 5 minutes before serving.