For the confit, remove any excess skin and fat from pieces of duck; cut skin and fat into cubes, place in a deep pot with 1 cup water, and slowly render fat over medium heat. When water has evaporated and remaining liquid is strained, clear duck fat will remain. Use the fat in this recipe, or pour it into a clean container, cover, and refrigerate or freeze for future use. Place the duck pieces in a large mixing bowl. Combine salt, pepper, thyme, rosemary, bay leaf, and quatre epices, and rub mixture over duck pieces. Cut unpeeled head of garlic in half crosswise, stud with cloves and add to bowl. Cover meat with plastic wrap, place a weight on it to press down on meat, and refrigerate for 1 day. (A bottle of Armagnac is a perfect weight unless you drink too much of it during the next 24 hours!) The next day, melt 8 cups duck fat in a large heavy casserole over medium heat. Remove duck pieces from refrigerator and wipe off herb and salt mixture. Reserve garlic studded with cloves. When fat is warm but not hot, add duck pieces to the pot along with clove studded garlic. Put in legs first, bringing to a low simmer. Half an hour later, add gizzards; then cook slowly over low heat for another 45 minutes. Simmer until meat can be easily pierced with a fork. Remove pieces from pot and set aside.;
For the garlic puree, thinly slice garlic and slowly saute in the 2 tablespoons of olive oil over low heat until lightly caramelized, 4 to 5 minutes. Use oil and garlic when you puree mixture.
For the succotash, heat duck fat in a large skillet over medium heat. Add onion, celery, and bell peppers and saute until softened but still bright in color, about 4 minutes. Add garlic, coriander, red pepper flakes, cumin, and bay leaf, and cook 1 minute. Stir in molasses, lima beans, and chicken stock, bring to a simmer, and cook until beans are done but still firm. Stir in corn, cook for 2 minutes, then season with salt and pepper. If not using immediately, cool quickly over ice to avoid overcooking. Cover and refrigerate until needed. Recipe may be prepared up to 2 days ahead to this point. Reheat over low heat. Stir in tomatoes, adjusting amount to your taste. Add thyme leaves, vinegar, and butter. Taste for seasonings and keep warm.
For the smoked tomato puree, heat canned tomato puree in a small skillet until slightly reduced; add a little chopped chipolte pepper or to taste. For the ketchup, combine tomato puree, ketchup, vinegar, oil, mustard seeds, dissolved tamarind water, garlic puree, garlic chili paste, hot pepper sauce, and Worcestershire sauce in an electric blender, and pulse until blended. Season with salt, scrape into a clean container, cover and refrigerate until needed.;