Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets for 3 minutes until bright green and slightly tender. Drain and set aside.
In a medium skillet, melt the butter over medium heat. Add the chopped onion and minced garlic and sauté for 3-4 minutes until softened and fragrant. Remove from heat.
In a large mixing bowl, combine the cream of mushroom soup, sour cream, milk, salt, black pepper, and dried thyme. Stir until smooth.
Add the cooked rice, shredded chicken, blanched broccoli, sautéed onion and garlic mixture, and 1/2 cup of shredded cheddar cheese to the bowl. Mix everything together until well combined.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
In a small bowl, mix the panko breadcrumbs, grated Parmesan cheese, and remaining 1/2 cup shredded cheddar cheese.
Sprinkle the breadcrumb and cheese mixture evenly over the top of the casserole.
Bake uncovered in the preheated oven for 35-40 minutes until the casserole is bubbly and the topping is golden brown.
Remove from oven and let cool for 5 minutes before serving.