Cook the wild rice according to package directions until tender, then drain any excess water and set it aside.
Preheat the oven to 375 degrees and lightly grease a 9 by 9 inch baking dish.
Melt the butter in a large skillet over medium heat. Add the onion and cook until softened.
Stir in the mushrooms and garlic, and cook until the mushrooms release their liquid and begin to brown.
Sprinkle the flour over the vegetables and stir for one minute to form a roux.
Slowly whisk in the chicken broth and milk, then add the thyme, salt, and pepper. Continue stirring until the sauce thickens.
Remove the skillet from the heat and stir in the cooked wild rice, diced turkey, and half of the cheddar cheese until evenly combined.
Transfer the mixture to the prepared baking dish and spread it into an even layer.
Top with the remaining cheddar cheese and the breadcrumbs.
Bake for 25 to 30 minutes until the top is golden and the edges are bubbling. Let it rest for five minutes before serving.