Preheat the oven to 350°F (175°C). Lightly 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 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 5-6 minutes until lightly browned but not fully cooked through. Remove chicken from skillet and set aside.
In the same skillet, add the chopped onion and cook for 3-4 minutes until softened. Add the minced garlic and sliced mushrooms and cook for another 5 minutes until mushrooms release their moisture and soften. 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 the mixture is 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 the sour cream, dried thyme, salt, black pepper, and smoked paprika until smooth.
In a large mixing bowl, combine the cooked wild rice, sautéed vegetables, browned chicken pieces, and creamy sauce. Stir until evenly mixed.
Transfer the mixture to the prepared baking dish. Sprinkle the shredded cheddar cheese evenly over the top.
Cover the dish with aluminum foil and bake in the preheated oven for 30 minutes.
Remove the foil and bake an additional 15 minutes until the cheese is melted and bubbly and the casserole is heated through.
Remove from oven and let rest for 5 minutes. Garnish with chopped fresh parsley before serving.