Preheat oven to 400°F and line a baking sheet with parchment paper.
Rinse the hajiki in a colander to remove any dirt, then place it in a bowl with enough warm water to cover it. Let it soak for 15 to 20 minutes, until it has reconstituted. Drain and coarsely chop into small pieces.
Add the vegetable oil to a small pan over medium-high heat. Add the carrot, scallion, garlic, sesame seeds and ginger. Cook, stirring frequently, for about 5 minutes or until the vegetables have begun to soften and are fragrant.
In a large bowl, mash the tofu with a potato masher or large spoon. Add the hajiki, cooked carrot mixture, miso, chickpea flour, tamari, sesame oil to the tofu and mix well. Roll up your sleeves and use your hands to get everything well mixed.
Form the tofu mixture into 4 patties and gently place them on the parchment paper. Cook for 15 minutes, flip the burgers, and cook for another 15 minutes, or until golden brown on both sides.
While the burgers are cooking, mix all of the ingredients together in a food processor or blender. It it seems too thick, add a little water, a teaspoon at a time. Set aside until ready to use.
Serve burgers on a plate with brown rice and a salad or on a burger bun with lettuce, tomato, and avocado. Drizzle the Ginger Miso Dressing generously over the burgers.