Preheat oven to 400F.
Add 1 cup of the pistachios to the bowl of a food processor and pulse until the nuts are coarsely ground. Set the grounds aside. Place the rest of the pistachios in the food processor and blend, slowly adding extra virgin olive oil until the mixture is the consistency of a paste. Add salt and pepper to taste.
Cut the two plums in half then remove and discard the pits. Place the plums in a blender, add the sherry vinegar and blend until smooth. Add salt and pepper to taste.
In a deep baking pan, pour enough kosher salt to cover the surface. Place the sunchokes in the pan then add enough salt so that the sunchokes are entirely covered. Cover the pan with aluminum foil and roast for two hours. Allow to cool, then remove the sunchokes, dusting off any remaining salt with a dry paper towel. Discard the salt. Slice the sunchokes into 1/4-inch slices. In a sauté pan over medium heat, add extra virgin olive oil until warm. Lightly season the sunchokes with salt and pepper and sauté until golden brown on both sides. Remove then dry on a paper towel.
Separate the legs and breasts of the duck. Pull off any loose fat and set it aside. (Make sure to save the carcasses for stock.)
In a small bowl, combine the red pepper flakes and 1 tablespoon each of salt and pepper. Rub the legs all over with the seasoning. Wrap the legs individually in plastic wrap, and refrigerate overnight along with the unseasoned breasts and the reserved duck fat.
In a small saucepan over low heat, combine the reserved duck fat and ¼ cup water and cook until the water has evaporated and the fat is liquefied, about 1 hour.
Preheat the oven to 325°F. Place the duck legs in a shallow, non-reactive casserole and cover with the hot fat. Cover and bake for 2 hours. Remove from the oven and allow to cool but keep the oven on. (Note: the legs may be stored in the refrigerator, covered in the fat, for up to 4 weeks.)
When the legs are cool, remove them from the fat and use a paper towel to wipe off the excess fat. Season the breasts and legs with salt and pepper. Then, using a sharp paring knife, score the breasts on the skin side with an X, being careful to penetrate the skin and fat but not the flesh.
In a 12- to 14-inch ovenproof sauté pan over high heat, heat 1 tablespoon of the olive oil until smoking. Add the legs and sauté them, skin side down, until the skin is crispy. Remove the pan from the heat and place it in the oven to finish cooking, about 5 minutes.
In a second sauté pan of equal size over low heat, add 1 tablespoon of olive oil. Place the seasoned breasts in the pan skin side down and sear for about 20 minutes, until crispy, draining the excess fat from the pan when necessary. Turn the breasts and cook on the flesh side until they are medium rare, or reach an internal temperature of 130F. Transfer the beasts to a cutting board and allow them to rest for 5 minutes. Slice them on a bias.
Place the Pistachio pesto on the bottom of a plate and cover with sunchokes. Carefully place the duck legs on top of the sunchokes. Fan the sliced duck breast over the legs. Drizzle the plum vinaigrette over the duck and sprinkle with ground pistachios. Finish with extra virgin olive oil and enjoy!