Preheat oven to 350°F.In a medium sauce pan over high heat, bring 2 cups water to a boil. Add wild rice, reduce heat to low to medium-low and cover. Let simmer for about 40 minutes, or until rice has split open (see picture in post). It may take an additional 10 to 15 minutes for this to happen. Drain and set aside.While wild rice is cooking, place mushroom caps on a rimmed pan and set aside. Place mashed potatoes in a medium bowl and set aside.In a fry pan over medium heat, melt butter. Add garlic and onion and cook until softened, about 7 to 8 minutes, stirring regularly. Do not let the garlic and onion brown. Remove pan from heat.Add softened garlic and onion, thyme, rosemary, salt, and pepper to mashed potatoes. Stir to combine. Then stir in 1 1/4 cup of the cooked wild rice, white cheddar, and Parmesan. Taste mixture, adding more salt and pepper if needed. With a spoon, add mashed potato mixture to cavities in mushrooms, stuffing the potato mixture in as tightly as you can. Then mound remaining potato mixture on top, pressing it onto the mushroom. Depending on the size of your mushroom caps, you may have less or more potato mixture to mound on top. Place stuffed mushrooms in oven and bake for 25 minutes. Add a little white cheddar, then Parmesan, on top of mashed potato mixture and return to oven to bake for another 10 to 15 minutes longer. Sprinkle with additional fresh chopped thyme and serve hot.