Preheat the oven to low to keep the duck warm later. Dry-fry the duck breasts, skin side down, in a large pan over moderate heat for about 8 minutes, then turn and cook for a further 6 minutes.
While the duck is cooking, peel the fresh ginger and cut into thin strips. Chop the stem ginger and put both aside.
Trim the leeks, cut in half lengthways, rinse and drain well, then cut into matchstick-sized strips and put aside.
Transfer the cooked duck to the oven to keep warm. Drain off all but about 1 tablespoon of the duck fat. Add the vinegar and bring to the boil, scraping up the brown residue, then add the fresh and stem ginger, ginger syrup, ginger wine and stock. Bring back to the boil, reduce the heat and simmer for 8–10 minutes, until the liquid has reduced by half.
Meanwhile, melt the butter in a small frying pan, add the leeks, season with salt and pepper and stir-fry gently for 3–4 minutes until just tender.
Return the duck breasts to the ginger sauce, reheat for 2 minutes, then add salt and pepper to taste.
Arrange the leeks on a serving dish, place the duck breasts on top and spoon over the sauce.
If you would prefer a thicker sauce, dissolve 1 teaspoon arrowroot or corflour in 3 tablespoons cold water, then pour into the sauce and simmer for 2minutes before you return the duck to the pan.