Place the duck breast-side up on a cutting board. Remove and reserve the giblets and discard the liver. Break down the duck and score the breasts.
Puree the shallots, ginger and coriander in a mini food processor. Season the duck legs and the meat side of the breasts with salt, pepper and 2 tablespoons of the ginger mixture. Cover and refrigerate the duck and the remaining ginger mixture while you make the stock.
Make the duck stock: Trim the excess fat from the duck bones and chop into 6 to 8 pieces. Heat 2 tablespoons olive oil in a Dutch oven or large pot over medium-high heat; add the bones, reserved wings and giblets and brown, turning occasionally, about 25 minutes. Add the broth and enough water to cover the bones. Bring to a boil, then reduce to a simmer and cook, uncovered, about 3 hours, skimming as needed. Strain the duck stock and skim off any excess fat from the surface. (The stock can be made a day ahead. Just cover and refrigerate.)
Preheat the oven to 375 degrees F. Place the duck legs on a rack in a roasting pan with 1/4 inch water. Roast until brown and crisp, about 1 hour 15 minutes.
Meanwhile, heat the remaining 1 tablespoon olive oil in a medium saucepan over medium-high heat; add the reserved ginger mixture and cook, stirring, until browned, 5 to 7 minutes. Add the port and scrape up any browned bits from the pan with a wooden spoon. Boil until the mixture looks like wet sand. Add 3 cups of the duck stock and the bay leaf and simmer until the liquid reduces by about three-quarters, about 45 minutes. Strain into a separate saucepan. Add the figs and heat until plump, about 2 minutes. Whisk in the butter, season generously with salt and pepper and add the vinegar. Keep the sauce warm over low heat but do not boil.
Scrape the ginger mixture off the breasts. Heat a medium skillet over high heat. Place the breasts, skin-side down, in the skillet and cook until the fat begins to render and the skin is golden brown, 1 to 2 minutes. Pour off the fat. Reduce the heat to low and continue cooking, removing the fat as it renders, until the skin is tight and golden, 15 to 20 minutes. Increase the heat to medium-high, flip the breasts and cook until the meat is lightly browned but still medium-rare, 1 to 2 more minutes.
Thinly slice the breasts and cut the legs in half. Divide the meat among plates and top with the sauce and figs.