Preheat the oven to 375°F (190°C).
In a medium saucepan, combine the brown rice and chicken broth. Bring to a boil, then reduce heat to low, cover, and simmer for 40 minutes, or until rice is tender and liquid is absorbed.
While the rice cooks, melt the butter in a large skillet over medium heat.
Add the diced onion, carrots, and celery to the skillet and sauté for 5-7 minutes until vegetables are softened.
Add the minced garlic and cook for 1 minute until fragrant.
Sprinkle the flour over the vegetables and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the milk, stirring constantly until the mixture thickens, about 3-4 minutes.
Remove the skillet from heat and stir in the sour cream, dried thyme, salt, and black pepper.
Add the cooked brown rice, chicken pieces, and frozen peas to the skillet. Mix everything until well combined.
Transfer the mixture to a greased 9×13 inch baking dish and spread evenly.
Sprinkle the shredded cheddar cheese evenly over the top.
Cover the casserole with aluminum foil and bake in the preheated oven for 25 minutes.
Remove the foil and bake for an additional 10-15 minutes, or until the cheese is melted and bubbly and the chicken is cooked through (internal temperature 165°F/74°C).
Let the casserole rest for 5 minutes before serving.