Heat oil in a small pot over low heat. Add 1 teaspoon ginger and garlic; cook and stir until slightly browned, about 2 minutes. Stir in 1/4 cup tamari, 1/4 cup sake, mirin, brown sugar, and sesame seeds; simmer until glaze is slightly thickened, about 5 minutes.
Combine ground chicken, egg, green onions, carrot, bread crumbs, miso, 2 tablespoons ginger, potato starch, seaweed, 1 tablespoon tamari, 1 tablespoon sake, soy sauce, white sugar, and paprika in a large bowl; mix well into a thick paste, at least 5 minutes.
Preheat grill for medium heat and lightly oil the grate.
Roll ground chicken mixture into balls the size of your palm. Flatten them slightly into patties. Grill until well-browned, 3 to 4 minutes. Flip and continue grilling until browned on the second side, 2 to 3 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).