Preheat the oven to 350°F (175°C).
In a medium saucepan, bring the chicken broth to a boil. Add the rice, reduce heat to low, cover, and simmer for 15 minutes until the rice is cooked and liquid is absorbed.
While the rice cooks, heat olive oil in a large skillet over medium heat. Add the diced onion and minced garlic and sauté for 3-4 minutes until softened.
Add the chicken pieces to the skillet. Season with salt, black pepper, and dried thyme. Cook for 6-8 minutes until the chicken is no longer pink in the center.
Stir in the frozen mixed vegetables and cook for another 2 minutes.
In a large bowl, combine the cooked rice, chicken and vegetable mixture, condensed cream of mushroom soup, sour cream, and milk. Mix well until fully combined.
Transfer the mixture into a greased 9×13 inch casserole dish. Spread evenly.
Sprinkle the shredded cheddar cheese evenly over the top of the casserole.
Bake in the preheated oven for 25-30 minutes until the casserole is bubbly and the cheese is melted and golden brown.
Remove from the oven and let it rest for 5 minutes before serving.