Using a very sharp knife, lightly score the skin on each duck breast half in a crisscross pattern. Sprinkle them lightly on all sides with salt and pepper.
Heat a large skillet over medium-high until hot. Reduce the heat to medium and place the duck breasts, skin side down, in the skillet. Cook for 10 minutes, or until the skin looks very crispy.
When the duck skin is crisp, transfer the breasts to a plate. Pour off all but 2 teaspoons of the fat from the pan. Reserve it for another use, such as sauteing vegetables. Return the duck to the skillet, skin side up, and cook for another 3 to 5 minutes for medium-rare. Transfer the duck to a clean plate, skin side up. Cover it loosely with foil and let it rest while you make the sauce.
Juice 2½ of the oranges — you need about ½ cup of juice. Thinly slice the remaining half.
Without cleaning the skillet, return it to medium heat and add the shallots. Saute until they are golden, 2 to 3 minutes. Add the orange juice and simmer until reduced by half. Add the sherry vinegar and simmer 2 minutes. Add the chicken broth and the orange slices and simmer until slightly syrupy, or reduced by about 1/3.
Whisk the cornstarch mixture into the sauce. Simmer, whisking for 1 minute. Add the mustard and any juices that have collected on the plate holding the duck breasts. Season with salt and pepper.
Remove and discard the skin from the duck, if desired. Thinly slice the duck and arrange on 6 plates. Spoon some of the sauce with the orange slices over each portion, then sprinkle with chives.