Preheat the oven to 350°F (175°C).
Heat olive oil in a large skillet over medium heat. Add diced onion and cook for 3-4 minutes until softened.
Add minced garlic to the skillet and cook for 1 minute until fragrant.
Add sliced mushrooms and cook for 5 minutes until they release moisture and soften.
In a large mixing bowl, combine the cooked onion, garlic, mushrooms, cooked chicken, broccoli florets, cooked rice, cream of mushroom soup, sour cream, milk, dried thyme, salt, and black pepper. Stir well to combine.
Add half of the shredded cheddar cheese and half of the shredded mozzarella cheese to the mixture and stir to incorporate.
Transfer the mixture to a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the remaining cheddar and mozzarella cheese evenly over the top.
Sprinkle breadcrumbs evenly over the cheese layer.
Bake uncovered in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.