Prepare the duck: Pat the duck dry and place on a rimmed baking sheet, skin side up, uncovered, and refrigerate for 30 to 45 minutes. If you have time, you can leave it in the fridge for up to 3 hours (see Kitchen Tips).
Preheat the oven to 325°Remove the duck from the refrigerator and sprinkle with salt and pepper. Make a bouquet garni: cut a large piece of cheesecloth and place the bay leaves, peppercorns, and cardamom pods on it. Gather the edges together, wrap into a small bundle, tie with a long string of kitchen twine and set aside.
Heat 1 tablespoon canola oil or chicken fat in an enamel-coated, cast-iron or heavy-bottomed Dutch oven set over very high heat until the oil shimmers or the fat is melted and very hot. Carefully place the duck pieces into the pan in a single layer, skin-side down, working in batches if necessary (do not crowd the pan). Reduce the heat to medium-high and cook undisturbed for 9 to 10 minutes, until the skin is a deep golden brown. With a spoon, transfer 4 tablespoons of the duck fat to a bowl and set aside. With tongs, remove the par-cooked legs to a rimmed platter. Remove the skin from the duck and set aside (it will be crisped up later).
Add the minced shallots, cinnamon stick, and bouquet garni to the Dutch oven, reduce the heat to a simmer, and cook for 2 to 3 minutes, until the shallots are translucent and have browned at the edges. Add the stock and the wine, bring to a boil, and then reduce the heat to a simmer. Cook for 10 to 15 minutes, until the liquid is reduced in volume by half. Add the quince jelly and dried figs and stir until the jelly dissolves. Return the duck to the pot, cover with foil, carefully place the pot into the oven, and braise for 1 hour to 1 hour and 45 minutes, or until the meat falls off the bone.
While the duck is in the oven, make the purées: Preheat the oven to 375°F and spray 2 rimmed baking sheets with nonstick vegetable oil spray. Arrange the celery root and parsnips on one, dab on 1 tablespoon of the reserved duck fat, drizzle with 1 tablespoon of the oil, sprinkle with the cardamom, cinnamon, and half the salt, and toss lightly to coat. Bake for 35 to 40 minutes, or until the vegetables are cooked through and a sharp knife slides in and out with ease.
Place the carrots on the other prepared baking sheet, dab on the remaining 1 tablespoon duck fat, drizzle with the remaining 1 tablespoon olive oil, sprinkle with the remaining salt, and toss gently to coat. Place in the oven with the other vegetables and bake for 30 to 35 minutes, or until cooked through.
When the vegetables are done, remove both baking sheets from the oven. Transfer the celery root and parsnips to a blender or food processor along with ½ cup chicken broth, and scrape in any flavorful browned bits that are stuck to the pan. Process until smooth and creamy, adding more stock if needed, a little at a time, to achieve a nice, smooth consistency. Set aside.
Wipe the blender or processor clean and transfer the carrots to it, again scraping in any juices and browned bits that are stuck to the pan, along with the ¼ cup stock, the honey, and the vinegar. Process until smooth and creamy, again adding more stock if needed, a little at a time, to achieve a nice, smooth consistency. Set aside.
When the duck is done, remove the pot from the oven and transfer the duck to a clean platter. Remove the meat from the bones, and discard the bones and gristle. Cut or shred the meat into bite-sized pieces and set aside. There will be copious amounts of rendered duck fat—culinary gold (see Kitchen Tips for many uses for duck fat). Skim the fat (not the juicefrom the Dutch oven. Discard the bouquet garni and cinnamon stick, and cook the contents of the pot over high heat, uncovered, for about 5 minutes, until the liquid has thickened, is reduced in volume and easily coats a spoon. Return the duck to the Dutch oven, and stir gently. Reduce the heat to a simmer, cover, and set aside. 1Make the greens. Heat a sauté pan, and when hot, add the fat and heat through. Add the chard, salt, and pepper and cook, stirring, for 2 to 3 minutes, until wilted. Transfer the greens to a container and wipe the skillet clean.
Set the skillet over high heat until it is very very hot. Meanwhile, line a platter with paper towels and position it near the stove. Carefully place the reserved duck skin into the pan, fat-side down, and cook undisturbed until any residual fat has melted and the skin is beginning to crisp and fry. Turn over with tongs and do not worry if the pieces come apart a bit. When they are toasted to a deep brown (but not burneremove from the pan with tongs and set aside on the prepared platter.
To serve, divide the celery root and parsnip purée, the carrot purée, and the greens evenly between 6 dinner plates, swirling the purées attractively. Top with equal portions of duck meat and drizzle the sauce over each. Crumble or cut the crispy duck skin and scatter over each portion. Garnish with fresh figs and serve immediately.