Put the aburaage and plenty of water in a pot, and bring to a boil. Simmer for 20 minutes to drain the excess oil from the aburaage very well. If you don't drain the excess oil from the aburaage properly, it won't soak up all the flavors and be a little bland.
Drain in a colander. When it cools a little, fold into half, and press down with your palm to drain the excess water really well.
Put the dashi stock, brown sugar crystals, soy sauce, mirin and aburaage into a large pot. Cover with a small drop lid that sits right on top of the aburaage, and simmer over low heat for about 20 minutes until the sauce reduces to 1/3 its original volume.
After simmering, let it cool a little. Transfer to an airtight container, and leave overnight or more. After sitting overnight, cut in half, and open up the aburaage carefully. Don't squeeze out the liquid.
Make the sushi vinegar. Add the salt to the rice vinegar, then add the kombu and let soak for at least 30 minutes to bring out the kombu's flavor. The aburaage is sweet, so you don't need to add sugar to the sushi rice.
Transfer the freshly cooked rice into a handai (circular wooden vat-like container for rice) or large bowl, add the sushi vinegar, white sesame, and the sauce from cooking the aburaage over the rice. Fold in gently with a spatula while fanning the rice.
Let the sushi rice cool down until just warm to the touch. Make 20 rice balls, and stuff in the aburaage pouches carefully. It's quicker and easier if you make rice balls first before stuffing them in the aburaage.
Don't stuff too much sushi rice into the aburaage. Gently stuff the rice, and leave the bottom of the pouch (the part I'm pointing to in the photo) without any rice.
Brush the sauce left over from cooking the aburaage on the inari sushi for glazing. Lightly cover with plastic wrap, and let sit for at least 2 hours so flavors settle in. It tastes better after all the flavors have melded than when freshly made!
Garnish with pickled ginger and they are all done! It's best to eat after 4-5 hours!
Here's what the cross section looks like after 5 hours. It's difficult to see in this photo, but the flavors from the aburaage soaked into the sushi rice and made it more delicious.