Wipe the shiitake mushrooms clean of any dirt, then separate the stems from the caps. Roughly chop the stems and thinly slice the caps.
Combine the shiitake stems, kelp, and water in a large saucepan or pot. Place the pot on medium heat until the water just begins to simmer, then reduce the heat to low. Cook at a very low simmer for about 20 minutes. Remove the kelp from the pot and roughly chop once cool. Also ladle out 3/4 cup of the broth and reserve it in a separate bowl.
Increase the heat to medium high and bring the pot to a gentle boil. Add the shiitake caps, garlic, and ginger and cook until the mushrooms are tender, about 5-6 minutes.
Add the bok choy and green onions to the pot and cook until tender, about 2-3 minutes.
Meanwhile, combine the miso paste and reserved broth in a separate bowl. Whisk until the mixture is smooth and the miso has completely dissolved.
Add the chopped kelp back to the pot, then remove it from heat and stir in the miso mixture. Ladle the soup into bowls and enjoy.