First, make the broth: in a large saucepan, combine the dashi, soy sauce, sake, mirin, and salt. Heat through and set aside.
Place the fried tofu in a separate shallow saucepan with the mirin, soy sauce and 1/2 cup of water. Bring to the boil, then cover and simmer, turning occasionally, until the tofu has softened, plumbed up, and takes on a deeper brownish color – roughly ten minutes. Turn off the heat and leave the tofu in the saucepan to keep it warm.
Cook the kishimen according to the directions on the package, but take them out while still chewy. Add to the dashi broth and mix for a minute or two.
Pour the kishimen broth mixture evenly into four large soup bowls, distribute the abura-age between them, and top with generous mounds of bonito shavings. Eat at once.