Bring water to a simmer in a medium saucepan over medium-high. Remove from heat, and add mushrooms. Stir to submerge mushrooms in water, and let stand, undisturbed, until plump, about 20 minutes. Using a slotted spoon, transfer mushrooms to a small bowl. Pour mushroom broth into a medium bowl through a fine-mesh strainer lined with a piece of cheesecloth; discard cheesecloth, and reserve mushroom stock. Remove and discard stems from mushrooms. Coarsely chop mushrooms.
Heat oil in a large saucepan over medium-high. Add chopped mushrooms, and cook, stirring occasionally, until slightly browned, about 5 minutes. Add thyme, and cook, stirring constantly, until fragrant, about 1 minute. Add reserved mushroom stock and turkey bone broth; bring to a boil over high. Reduce heat to medium, and simmer, undisturbed, until reduced to 2 cups, about 20 minutes. Remove from heat; discard thyme sprigs. Whisk in miso, vinegar, salt, and pepper. Serve hot.