Preheat the oven to 350°F (175°C). Grease a 9×13 inch baking dish.
In a medium saucepan, bring the chicken broth to a boil. Add the rinsed wild rice, reduce heat to low, cover, and simmer for 45 minutes or until rice is tender and liquid is absorbed. Remove from heat and set aside.
While the rice cooks, melt the butter in a large skillet over medium heat. Add the diced onion, minced garlic, sliced mushrooms, diced celery, and diced carrots. Cook, stirring occasionally, until vegetables are tender, about 7-8 minutes.
Sprinkle the flour over the cooked vegetables in the skillet and stir well to coat. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in the milk, stirring constantly until the sauce thickens, about 3-5 minutes.
Remove the skillet from heat and stir in the sour cream, salt, black pepper, dried thyme, and dried parsley until well combined.
In a large mixing bowl, combine the cooked wild rice, cooked chicken, and the creamy vegetable sauce. Mix well.
Transfer the mixture to the prepared baking dish and spread evenly.
Sprinkle the shredded cheddar cheese and mozzarella cheese evenly over the top.
Sprinkle the crushed buttery crackers evenly over the cheese layer.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.