Bring the white vinegar, cider vinegar, brown rice syrup, mustard seeds, salt, and ¾ cup water to a boil in a small pot over high heat. Stir the radish and leeks into the boiling liquid, cover the pot, and turn off the heat. Allow the vegetables to marinate in the covered pot for 10 minutes.

Pour the pickled vegetables and brine into a glass jar and cool, without the lid, for at least 2 hours in the refrigerator. Use immediately, or store for up to 10 days, covered with a lid, in the refrigerator.

Bring the agave nectar, cider vinegar, and 1 cup water to a boil in a small saucepan over high heat

Place the Korean chile, rice flour, soy flour, all-purpose flour, and salt in a stand mixer fitted with the paddle attachment. Mix the ingredients at low-medium speed, while slowly pouring in the hot liquid ingredients. Continue to mix for about 2 minutes to form a sticky red paste. Allow the paste to cool, and use immediately; or store in an airtight container in the refrigerator for up to 2 weeks.

Preheat the oven to 350°F. Line a rimmed baking sheet with parchment paper and lightly oil the paper.

Place the seitan, onions, coriander, red pepper flakes, Korean chile, tomato paste, soy sauce, and olive oil in a food processor and process for 2 minutes, or until the seitan and onions are in very small fragments. Place the ground mixture in a mixing bowl and thoroughly mix the flour in.

Divide the mixture into 4 equal portions. With wet hands, roll the portions into balls and flatten them gently into smooth disks on the prepared baking sheet. Bake the burgers for 12 to 14 minutes, until the outsides of the burgers are firm but not dried out.

Heat the sesame oil in a medium sauté pan over medium heat. Sauté the kimchi in the hot oil for 5 minutes, until hot and fragrant.

In a separate pan over medium heat, melt the margarine. Cook the burgers on one side for 2 minutes. Flip the burgers and cook for 2 minutes on the second side.

Split the burger buns and toast in a third pan, cut side down. Spread 2 tablespoons of the Sriracha mayonnaise onto the cut sides of each bun. Place ¼ cup of the arugula on the bottom half of each bun, followed by a burger. Spread 1 to 2 tablespoons of the gochujang on each patty, spoon on ¼ cup of the sautéed kimchi and 2 to 3 tablespoons of the pickled vegetables, and then add 1 tablespoon of the gomasio. Finish the burger with the top half of the bun.