Preheat your oven to 375°F (190°C).In a large skillet, melt the butter over medium heat. Add the chopped onion and garlic, sautéing until they become translucent and fragrant. Stir in the flour, dried thyme, dried parsley, salt, and pepper. Cook for another minute to create a roux.
Gradually pour in the chicken broth and milk, stirring constantly to avoid lumps. Let the mixture simmer for about 2 minutes until it thickens.
Add the shredded chicken, chopped carrots, and frozen peas to the skillet, stirring until all ingredients are well combined.
In your casserole dish, spread the uncooked rice evenly on the bottom.Pour the chicken and vegetable mixture over the rice layer in the casserole dish. Cover the dish with aluminum foil and bake in the preheated oven for 30 minutes.
After 30 minutes, remove the foil and bake for an additional 10 minutes to let the top layer get a golden-brown color.
Once the casserole is cooked, remove it from the oven, and let it cool for a few minutes before serving.