Duck breasts: Sprinkle each duck breast liberally with salt and pepper. Heat the vegetable oil in a large cast-iron skillet over medium-low heat. Add the duck skin-side down and reduce the heat to low, cooking as the fat slowly renders and the skin becomes crispy, 8 to 10 minutes.
Once the skin is crispy and golden brown, flip and continue cooking until a thermometer reads 128 to 130 degrees F when inserted into the thickest part of the breast (for medium-rare doneness), 5 minutes. Transfer to a plate or cutting board and allow to rest about 5 minutes. It will carryover cook to about 135 degrees F. Don't tent with foil in order to ensure the duck skin will stay crispy.
For the fig sauce: Pour off all but 1 or 2 tablespoons of fat from the skillet, reserving the excess for another use. Over medium heat, add the shallots and cook until softened. Add the sherry and reduce by half. Next, add the chicken broth, fig jam and balsamic vinegar, and continue simmering until the sauce has thickened and is syrupy, another 5 to 7 minutes. Remove from the heat, season with salt and pepper and whisk in the butter. Garnish with chopped chives.
Serve the sauce alongside thinly sliced duck breast.