How to Make It
In a small bowl, cover the dried mushrooms with 2 cups of very hot water. Let soak until softened, about 20 minutes. Drain the mushrooms, reserving the soaking liquid. Rinse the mushrooms and finely chop them, discarding any tough bits.
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.
Strain the stock, pressing down on the solids. Skim off any fat. Wipe out the saucepan and pour in the stock. Boil the stock over high heat until reduced to 1 1/2 cups.
Heat the olive oil in a large non-stick skillet until smoking. Add the sliced shiitake and white mushroom caps and cook over high heat, stirring occasionally, until well browned, about 12 minutes. Stir the sliced mushrooms into the reduced stock and bring to a simmer over moderate heat. In a small bowl, mix the arrowroot with 2 tablespoons of water and stir the mixture into the gravy. Cook, stirring, just until thickened, about 3 minutes; do not let the gravy boil. Season with salt and pepper. Stir the chives into the gravy and serve.