Put water and kombu in a pot and let it sit for at least 30 minutes until the kombu softens.
Heat the pot over very low heat for about 10 minutes (for 2 servings). When small bubbles appear on the surface of the liquid, remove the kombu.
Increase the heat and bring the water (kombu dashi) to a boil. Once it comes to a boil, reduce the heat to low, add the bonito flakes, and let it simmer for 2 minutes.
Turn off the heat and strain the mixture through a sieve lined with paper towels or cloth (such as cheesecloth). Awase dashi, which combines kombu dashi and bonito dashi, is ready.
Cut the cabbage into bite-sized pieces. Thinly slice the green onions.
Return the dashi to the pot and bring it to a boil. Once it comes to a boil, gently crack the eggs into the pot. Let them simmer over low heat for 3 minutes.
Add the cabbage to the pot and continue cooking for about another 30 seconds.
Turn off the heat and dissolve the miso in the pot using a ladle and chopsticks, being careful not to break the eggs. Serve in each bowl and top with the green onions.