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For the ketchup leather: Combine the tomato paste, sugar, tomatoes, salt, onion powder, soy sauce, allspice, garlic, mushroom powder and black pepper with 3 1/4 cups water in a large pot and blend with an immersion blender until smooth. Dehydrate in a dehydrator according to manufacturer's instructions at 135 degrees F until dry, pliable and no longer wet to touch, 3 to 4 hours. Portion into 4-inch squares and store in wax paper.
For the Americanized dashi cheese: Soften the gelatin in a little milk in a small bowl. Make dashi milk broth by soaking the kombu in the milk for 1 hour and then simmering for 15 minutes. Turn off the heat and add the kastuobushi. Let the mixture sit for 10 minutes, then strain out the kombu and katsuobushi. Bring the dashi milk broth back to a simmer and whisk in the sodium citrate. Add the Cheddar and Parmesan in 3 stages, mixing to melt them between each. Add the softened gelatin and mix well. Pour onto a sheet tray lined with wax paper. Chill until the gelatin sets, then cut into 4-inch squares.
For the schmaltz onions: Render the chicken skin in a large skillet over medium heat until crispy. Remove the skin but leave the fat in the pan. Add the onions to the fat, then add the salt and pepper and caramelize the onions slowly until deep brown and sweet, about 2 hours. Chop up the chicken skin into tiny pieces and add it to the onions. Add the Worcestershire and adjust seasoning.
For the crunch bun: Heat a griddle to medium heat. Combine the butter, Parmesan and truffle salt in a small bowl. Place the panko on a small plate. Spread the butter mixture on top of the bun, dip the bun in the panko and toast on the griddle until toasted and warm.
For the beef patty: Gently form a patty with the ground beef using a 4-inch ring mold. Season both sides with salt and pepper. Cook in a cast-iron pan over medium-high heat until a crust forms and the patty is medium rare, about 130 degrees F. Top the cooked patty with 1 piece of Americanized dashi cheese and let it melt. Top that with 1 tablespoon of warmed schmaltz onions, then the pickle slices. Place 1 piece of ketchup leather on the bottom of the crunch bun. Add the patty and the top of the bun. Reserve the remaining ketchup leather, Americanized dashi cheese and onions for other burgers.