With the point of a knife, score the skin side of the duck breasts in crosshatch pattern, being careful not to pierce the flesh of the meat.
Season the duck breasts with salt and pepper. Heat a saute pan over medium heat for 2 minutes before adding the breasts, skin side down. Cook over medium to low heat for approximately 6 minutes to 8 minutes to render the fat.
Before turning the breasts over, carefully remove the excess fat from the saute pan. Turn the breasts over and saute the flesh side for 3 to 4 minutes before removing the duck breasts from the pan.
Add the shallots to the pan in which the duck breasts were cooked and return to the heat. Saute the shallots briefly before adding the ginger, add the honey and mirin, stir to combine and add the hot broth. Reduce to a simmer and cook until half the liquid has evaporated. Add the soy and cabbage and cook for 2 minutes before adding the scallions, stir, and remove from the heat.
The breasts have rested for several minutes and should now be medium rare. They can be sliced lengthwise or crosswise and several slices placed on each plate. Quickly reheat the sauce, if necessary, and drizzle some around the duck breasts.