Begin by preparing the vegetables. Dice the onions, celery, and carrots, and slice the mushrooms.
In a saucepan, melt the butter over medium heat. Add the flour and whisk continuously until it forms a roux, about 2-3 minutes.
Stir in the diced onions, celery, and carrots. Cook until they begin to soften.
Season the vegetable mixture with garlic powder, dried thyme, salt, and pepper. Pour in the chicken broth and milk
In a large mixing bowl, combine the cooked and shredded chicken with the cooked rice.
Pour the vegetable and sauce mixture over the chicken and rice. Mix until everything is evenly coated.
Grease a casserole dish and transfer the chicken and rice mixture into it.