Warm a large deep skillet over high heat. Add the chicken wings and cook, turning once, until well browned, about 15 minutes. Transfer the wings to a plate and pour off all the fat from the skillet. Add the reconstituted dried mushrooms and the shiitake and white mushroom stems and cook, stirring occasionally, until beginning to brown, about 4 minutes. Add the brandy and boil until evaporated. Pour in the mushroom soaking liquid, stopping when you reach the grit. Bring to a boil, scraping up any browned bits. Add 4 cups of water, the chicken wings, carrots, shallots, thyme, bay leaf, and peppercorns to the skillet and bring back to a boil. Cover partially, reduce the heat to moderately low, and simmer the stock for 1 1/2 hours.