Preheat oven to 500°F with rack in middle. Pat chicken dry, then roast, skin side up, in 1 layer in a 17- by 12-inch shallow baking pan until skin is golden brown, 35 to 40 minutes. While chicken roasts, soak wood ear mushrooms in 4 cups water until softened, about 15 minutes. Drain in a sieve, then rinse well and discard any hard pieces. Drain well, squeezing out excess water. Transfer roasted chicken to a bowl and pour pan juices through a fine-mesh sieve into a 1-quart glass measure. Let stand until fat rises to top, 1 to 2 minutes, then skim off and discard fat. Add enough stock to bring total to 4 cups liquid. Reduce oven to 300°F and move rack to lower third. Peel burdock root, and, if more than 1-inch-thick, halve lengthwise. Cut crosswise into 1-inch pieces. Transfer burdock root to a bowl, then add vinegar and 2 cups water. Heat oil in a 7- to 8-quart heavy pot over medium-high heat until it shimmers, then saute onions until softened and beginning to brown. Add shiitakes, ginger, and garlic and saute until garlic is golden, 3 to 5 minutes. Add mirin and boil, stirring and scraping up any brown bits, 1 minute. Stir in miso and soy sauce, then stir in chicken, wood ear mushrooms, burdock (drained), stock mixture, and remaining 4 cups water. Bring to a boil, skimming off any froth. Cover pot and braise in oven until chicken is tender, about 1 hour. Stir in mustard greens and continue to braise, covered, 5 minutes. Serve in shallow bowls.