Be sure to use ripe Hachiya persimmons, which are super soft to the touch. Otherwise, they will be too astringent for this sweet and tangy salsa. More Incredible Duck Recipes
Preheat the oven to 400°. In a medium bowl, toss the persimmons with the shallot, red wine vinegar and olive oil. Season with salt and pepper. Let the salsa stand at room temperature for 30 minutes.
Using a sharp knife, score the duck skin in a crosshatch pattern. Season the duck breasts with salt and pepper. In a large ovenproof skillet, cook the duck breasts skin side down over moderate heat until the fat is rendered and the skin is deep mahogany in color, about 15 minutes. Turn the breasts skin side up and transfer the skillet to the oven. Roast the duck for about 8 minutes for medium rare meat. Transfer the duck breasts to a carving board and let stand for 5 minutes.
Slice the duck on the bias 1/4 inch thick and transfer to plates. Serve with the persimmon salsa.