Dry the duck breasts very well. Trim any silverskin from the meat side of the breasts. If there’s a lot of skin, trim the edges so there’s only about 1/4 inch overhang. With a very sharp knife, score the skin in a 1/2-inch crosshatch pattern being careful to cut only through the skin and not into the meat. Season the breasts well on both sides with salt and pepper.

Put a tablespoon of olive oil in a large, heavy-bottomed sauté pan (don't preheat it!). Arrange the duck breasts, skin side down, in the cold pan. Be sure that the duck fits loosely in the pan and is not smushed or overlapping. Turn the heat on low and cook very slowly to render the fat from the duck breasts. It can take about 30-40 minutes to do this but will result in very crispy duck skin. As the fat builds up in the pan, remove it with a spoon or a baster, but reserve it (to cook potatoes or cipollini onions in).

When the fat has fully rendered from the duck breasts, turn the heat up to medium-high and cook until the skin is crisp and deep golden brown. Flip and brown the other side, another 2 to 4 minutes. To check for doneness, use an instant-read thermometer: 135°F for medium rare, 145°F for medium. Transfer the duck to a plate or platter and tent loosely with foil to keep warm. Make sure the duck rests for at least 10 minutes before slicing.

Remove any leftover fat in the pan and put back on medium-high heat. Add in the sherry, sherry vinegar and the figs.Scrape the bottom of the pan with a wooden spoon to release any cooked-on bits. Boil until the liquid has reduced to 1/4 cup, 3-5 minutes. Add the broth, honey, thyme, and 1/4 tsp. salt. Boil until the liquid is reduced by about half and the figs are soft, about 5 minutes (this part can be done in advance in a saucepan and then added to the pan in which the duck was cooked. Don't add the butter until just before serving). Reduce the heat to low and, while stirring constantly, add the butter into the sauce.

Slice the duck thinly on an angle, arrange it on plates, and spoon the sauce and figs around it to keep the skin crispy. Serve extra sauce on the side.