Preheat the oven to 350°F (175°C).
In a medium saucepan, combine the rinsed wild rice and chicken broth. Bring to a boil over medium-high heat, 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, heat olive oil in a large skillet over medium heat. Add the chopped onion and sauté for 3-4 minutes until softened.
Add the minced garlic and sliced mushrooms to the skillet. Cook for 6-8 minutes, stirring occasionally, until mushrooms are browned and moisture has evaporated. Remove from heat.
In a separate medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the whole milk, continuing to whisk until the sauce thickens and is smooth, about 4-5 minutes.
Remove the sauce from heat and stir in the sour cream, dried thyme, salt, and black pepper until well combined.
In a large mixing bowl, combine the cooked wild rice, mushroom mixture, sauce, and raw chicken pieces. Stir until everything is evenly incorporated.
Transfer the mixture to a greased 9×13-inch baking dish. Sprinkle the shredded cheddar cheese evenly over the top.
Cover the casserole with aluminum foil and bake in the preheated oven for 45 minutes.
Remove the foil and bake for an additional 15-20 minutes, or until the chicken is cooked through (internal temperature reaches 165°F/75°C) and the cheese is golden and bubbly.
Remove from the oven and let the casserole rest for 5 minutes.
Sprinkle the chopped fresh parsley over the casserole before serving.