1. Bring 2 cups chicken stock to a boil in a medium saucepan. Place dried porcini and morel mushrooms in a medium heat-proof bowl. Pour chicken stock over mushrooms; let steep 20 minutes. Using a slotted spoon, transfer mushrooms to a plate. Pass soaking liquid through a sieve lined with a double layer of cheesecloth set over a bowl. Reserve liquid.

2. Heat 1 tablespoon butter and olive oil in a medium stockpot set over medium-low heat. Add onion and garlic; sauté until translucent, about 7 minutes. Add all reserved porcini and morel mushrooms, assorted wild mushrooms, and rosemary and thyme sprigs; season with salt and pepper. Raise heat to medium high; cook until mushrooms have softened and released their liquid, about 5 minutes.

3. Add remaining 4 cups chicken stock and reserved mushroom liquid; bring to a boil. Partially cover, reduce heat to medium low, and simmer 30 minutes.

4. Remove and discard the herb sprigs. Working in batches, transfer the soup to the bowl of a food processor or blender, and puree until smooth.

5. Return the soup to the stockpot. Add the cream and milk; bring just to a boil. Season with salt and pepper.

6. In a small skillet, melt the remaining tablespoon butter. Add Hon Shimeji mushrooms, if using; cook over medium-high heat until golden brown, about 6 minutes.

7. When ready to serve, divide the soup among 8 soup plates, being sure to distribute the mushrooms evenly. Using a vegetable peeler or a truffle shaver, thinly shave the white truffles over each serving.