To make dashi, pour the water into a large stock pot and add the kombu. Bring to a gentle boil, then add katsuobushi and remove from heat. Let the dashi stand for at least 15 minutes, then strain out the solids and reserve the liquid.
(See note for a vegetarian version)
Add the tamari, mirin, miso, and sesame paste to the dashi stock. Whisk together to ensure that the miso is broken up and evenly distributed.
Bring the stock to a gentle boil and add the negi, chinese cabbage, shimeji mushrooms, and fried tofu. The tofu will float, but you can try to push it into the liquid to some extent. Reduce the heat, cover, and simmer for 10 minutes. Alternately, if you want hard-boiled quail eggs, simmer for 5 minutes, add the eggs, and simmer for an additional 5 minutes before continuing on to the next step (see also step 5 below). You ma
Add the chrysanthemum greens, enoki, and shiritaki noodles, push the tofu puffs back into the broth as best as you can, then cover and simmer for 3-4 minutes.
Serve immediately. If you haven't hard boiled the quail eggs, you can crack them into steaming-hot individual bowls at the table so that they partially cook in the hot soup.
Notes
Traditional Japanese stocks generally contain katsuobushi, which is made from skipjack tuna. If you want to make this dish vegetarian, simply omit the katsuobushi or add the liquid from re-hydrating 3-4 dried shiitake mushrooms in its place.To convert this into a vegan dish, simply follow the vegetarian instructions and omit the eggs.