Preheat your oven to 350°F (175°C). Grease a 9x13-inch baking dish.

If not already cooked, prepare the wild rice according to the package instructions. Wild rice takes about 45 minutes to cook.

In a large skillet, melt butter over medium heat. Add onion, garlic, carrots, celery, and mushrooms. Sauté for 5-7 minutes until softened. Season with salt, pepper, and thyme.

In a bowl, whisk together the cream of mushroom soup, cream of chicken soup, chicken broth, and milk. Stir in shredded cheese until well combined.

In a greased baking dish, combine the cooked wild rice, shredded chicken, sautéed vegetables, and creamy sauce. Stir until well-mixed.

In a small bowl, mix panko breadcrumbs, melted butter, and Parmesan cheese. Sprinkle this mixture evenly over the casserole.

Bake uncovered for 30-35 minutes, or until golden and bubbly. Let the casserole rest for 5 minutes before serving.