01 - Preheat oven to 375°F. Thoroughly grease a 9x13-inch casserole dish.
02 - Rinse wild rice under cold water. In a medium saucepan, combine the rinsed wild rice and vegetable broth. Bring to a boil over high heat, then reduce heat to low, cover, and simmer for 40 to 45 minutes, or until the rice is tender. Drain any excess liquid if necessary.
03 - While the rice cooks, heat 2 tablespoons of olive oil in a large skillet over medium heat. Add the finely chopped yellow onion and sauté until translucent, about 4 minutes. Incorporate the minced garlic, sliced mushrooms, diced celery, and diced carrot. Continue to cook for 8 to 10 minutes, stirring occasionally, until the mushrooms are well browned and the vegetables are softened. Stir in the fresh thyme, dried oregano, salt, and black pepper. Remove the skillet from the heat.
04 - In a separate medium saucepan, melt 4 tablespoons of unsalted butter over medium-low heat. Whisk in 1/4 cup all-purpose flour and cook for 1 minute, forming a roux. Gradually add 2 cups of whole milk and 1/2 cup of heavy cream, whisking continuously to prevent lumps, until the sauce is smooth and thickened, approximately 4 to 5 minutes. Stir in 1/2 cup of grated Parmesan cheese until it is fully melted and incorporated.
05 - In a large mixing bowl, combine the cooked wild rice, the sautéed vegetable mixture, and the prepared creamy sauce. Mix thoroughly until all components are evenly distributed.
06 - Transfer and spread the combined mixture evenly into the prepared 9x13-inch casserole dish. Sprinkle 1 cup of shredded Gruyère or Swiss cheese evenly over the top. For an optional crunchy topping, toss 1/2 cup of panko breadcrumbs with 1 tablespoon of melted butter, then scatter them over the cheese.
07 - Bake the casserole uncovered in the preheated oven for 25 to 30 minutes, or until the topping is golden brown and the filling is bubbling hot. Allow the casserole to rest for 10 minutes before serving to set.