Preparation

• In a medium bowl, coat the duck legs with 1 cup of the miso. Cover and refrigerate for at least 12 hours and up to 2 days.
• Position a rack in the center of the oven and heat the oven to 250°F.
• Rinse the miso from the duck legs and pat dry. Arrange the legs skin side down in a wide 7-quart heavy-duty pot, and cook over medium heat until the skin is golden brown and the fat has rendered, 5 to 7 minutes.
• Transfer the duck to a plate, discard or save the fat, and return the duck to the pot, skin side up. Add the shallots, carrots, sake, mirin, rosemary, ginger, and pepper flakes, tucking the vegetables around the legs. Pour in enough broth to barely cover the contents of the pot. Bring to a boil and skim off any foam from the surface. Cover and braise in the oven until the meat is fork tender, 1-1/2 to 2 hours.
• Transfer the duck and vegetables to a platter and keep warm, covered with foil. Pour the braising liquid into a fat separator and let sit about 5 minutes for the fat to rise. Discard or reserve the fat and return the liquid to the pot. Boil the liquid until reduced by half (about 1-1/2 cups), about 10 minutes. Whisk the remaining 1 Tbs. miso into the liquid. Pour some jus over the duck and vegetables, garnish with the scallions, and serve, passing any remaining jus at the table.