Put the washed rice in an electric rice cooker or large saucepan, add the sake and slightly less water than usual (the aim is firm rice, not mushy), top with the kelp, and cook. Meanwhile, slowly heat the vinegar, sugar, and salt, stirring continuously, in another pan. When the sugar is dissolved, remove from heat and let cool to room temperature.
As soon as the rice is ready, spread it on a large platter and cool quickly using a paper fan. Discard the kelp. Drizzle the cooled vinegar mixture over the rice and mix lightly with a rice paddle – trying not to mash the rice. Sprinkle in the sesame seeds and mix lightly. Leave to cool, covered with a damp piece of cheesecloth, until ready to use.
Place the abura-age in a colander and pour boiling water over it to remove the oil, and soften the abura-age. Gently pat dry with paper towels and slice in half (across the middle, to preserve two pouch-like ends). Carefully open the pouches with your fingers, trying not to tear them.
In a medium-sized saucepan, heat all the ingredients for the seasoning stock. Add the abura-age and bring to a boil. Turn the heat down and simmer, semi-covered, for 20 minutes. Remove from heat, leaving the abura-age in the sauce to absorb the flavor until it cools to room temperature.
Drain the abura-age, discarding the stock. Wet your hands with water and scoop up some of the sushi rice into a small ball, shaping firmly to fit one of the pouches. Stuff the pouches and place on a serving plate, seam side down.