Bring a pot of water to boil over medium heat. Carefully add eggs to the boiling water and cover. While eggs are boiling, prepare a large bowl of ice water. Cook for 6 minutes, then immediately remove eggs and place into prepared ice bath. Let cool completely.
To make the ramen broth, heat a thin layer of oil in a wok, stock pot, or Dutch oven over medium heat. Add the whites of the scallions, ginger, and garlic and cook gently until softened, about 3 minutes. Do not allow to burn.
Add sliced mushrooms and continue to cook until soft and dark, about 3-4 minutes longer.
Add mirin to the pot and scrape any browning bits from the bottom. Simmer gently until mushrooms are deeply browned and glossy.
Add the miso and soy sauce and combine completely. The vegetables should be a deep brown color.
Pour in the broth and stir well to incorporate. Bring to a simmer, then add the baby bok choy. Simmer until bok choy is softened and bright green in color, about 4-5 minutes.
Once bok choy is softened, remove from the heat. Divide the cooked noodles into two bowls and ladle broth evenly over noodles, adding the mushrooms and bok choy to the top of each bowl.
Gently crack the egg shells and peel away. Slice in half and add to each bowl of ramen.
Sprinkle with sesame seeds and the reserved green slices of the scallions to serve.