Roll the Aburaage with a rolling pin a couple of times to flatten them. Cut one side of each Aburaage. Carefully open each Aburaage to make a pouch. Boil the prepared Aburaage in hot water for a minute and drain well.
Place water, soy sauce, and sugar in a pot and boil. Add the Aburaage, cook for 10-15 minutes, and let cool for at least 3 hours (overnight in the fridge).
Hydrate shiitake mushrooms in water for 30 minutes. You can reserve the liquid to mix with Dashi later if you want.
Remove stems from Shiitake mushrooms and slice thinly. Cut carrots thinly (1 1/2" long).
Boil Dashi with soy sauce and sugar in a pot, add carrots and mushrooms until the liquid is almost gone.
Mix sushi rice, cooked vegetables, and sesame seeds well. Make 8 small oblong rice balls (there may be leftover rice). Squeeze the Aburaage just a little and stuff rice balls in the pouches.