Preheat the oven to 350°F (175°C).
In a medium saucepan, combine the wild rice and chicken broth. Bring to a boil, then reduce heat to low, cover, and simmer for 45 minutes or until the 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 chopped onion and sliced mushrooms. Sauté for 5-7 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Sprinkle the flour over the vegetables in the skillet and stir constantly for 2 minutes to form a roux.
Gradually whisk in the milk, continuing to stir until the sauce thickens, about 3-5 minutes.
Remove the skillet from heat and stir in the sour cream, dried thyme, salt, and black pepper.
In a large mixing bowl, combine the cooked wild rice, shredded chicken, and mushroom sauce. Mix thoroughly.
Stir in half of the shredded cheddar cheese and all of the shredded mozzarella cheese.
Transfer the mixture to a greased 9×13-inch casserole dish. Sprinkle the remaining cheddar cheese evenly on top.
Bake uncovered in the preheated oven for 25-30 minutes, or until the casserole is bubbly and the cheese is melted and golden.
Remove from the oven and let it rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.