Preheat the oven to 350°F (175°C).
In a large skillet, heat olive oil over medium heat. Add the chopped onion and minced garlic, and sauté for 3-4 minutes until softened.
Add the chicken pieces to the skillet and cook until lightly browned on all sides, about 5-6 minutes.
In a large mixing bowl, combine the rinsed rice, frozen peas, sliced mushrooms, cream of mushroom soup, cream of chicken soup, chicken broth, milk, dried thyme, salt, and black pepper. Stir well to combine.
Add the cooked chicken, sautéed onions, and garlic to the rice mixture and stir until evenly distributed.
Transfer the entire mixture into a greased 9×13 inch casserole dish and spread evenly.
Cover the casserole dish tightly with aluminum foil.
Bake in the preheated oven for 40 minutes.
Remove the foil and sprinkle the shredded cheddar cheese evenly over the top of the casserole.
Return the casserole to the oven uncovered and bake for an additional 5 minutes, or until the cheese is melted and bubbly.
Remove from oven and let the casserole rest for 5 minutes before serving.