Rinse the wild rice under cold water, then combine it with the vegetable broth in a saucepan. Bring to a boil, reduce heat, cover, and simmer for 45 minutes until the rice is tender and most of the liquid is absorbed.
While the rice cooks, melt the butter in a large skillet over medium heat. Add the diced onion and cook for 3 minutes until it softens.
Stir in the sliced mushrooms, minced garlic, and dried thyme. Cook for 5 to 6 minutes until the mushrooms release their liquid and begin to brown. Season with salt and pepper.
Preheat the oven to 375 degrees. In a large bowl, mix the cooked wild rice with the mushroom mixture, heavy cream, and half of the shredded cheddar cheese.
Transfer the mixture to a greased 8 by 8 inch baking dish. Sprinkle the remaining cheese evenly over the top.
Bake for 20 minutes until the cheese melts and the edges bubble. Remove from the oven and let it rest for 5 minutes before sprinkling with chopped parsley.