Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole 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 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-high heat. Add the chicken pieces and cook for 5-7 minutes until lightly browned and cooked through. Remove chicken from skillet and set aside.
In the same skillet, add the chopped onion and cook for 3 minutes until softened. Add the minced garlic and sliced mushrooms, cooking for an additional 5 minutes until mushrooms are tender and most liquid 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 and bubbly.
Gradually whisk in the whole milk, stirring constantly to avoid lumps. Continue cooking and stirring until the sauce thickens, about 5 minutes.
Remove the sauce from heat and stir in salt, black pepper, and dried thyme.
In a large mixing bowl, combine the cooked wild rice, cooked chicken, mushroom mixture, and creamy sauce. Stir until evenly combined.
Pour the mixture into the prepared casserole dish. Sprinkle shredded cheddar cheese and grated Parmesan evenly over the top.
Bake uncovered for 25-30 minutes until the casserole is bubbly and the cheese is melted and lightly golden.
Remove from oven and let rest for 5 minutes. Garnish with chopped fresh parsley before serving.