Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add the chopped onion, minced garlic, diced carrots, diced celery, and sliced mushrooms to the skillet. Sauté for 5-7 minutes until the vegetables are softened.
In a large mixing bowl, combine the cooked chicken breast, uncooked brown rice, sautéed vegetables, chicken broth, milk, Greek yogurt, dried thyme, dried parsley, black pepper, and salt. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Cover the casserole dish tightly with aluminum foil and bake in the preheated oven for 50 minutes.
Remove the foil and sprinkle the shredded cheddar cheese evenly over the top of the casserole.
Return the casserole to the oven and bake uncovered for an additional 10 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.