How to Make It
Heat oil in a large saucepan over medium-high. Add miso, onion, and ginger; sauté 10 minutes or until browned and fragrant. Add garlic; sauté 2 minutes. Add 4 cups water, stock, and mushrooms. Bring to a boil.
Carefully drop eggs into broth; reduce heat to medium. Simmer 7 minutes. Using a slotted spoon, transfer eggs to a bowl filled with ice water. Let stand 1 minute. Peel eggs.
Add buckwheat and carrot to broth; increase heat to medium-high, and bring to a boil. Reduce heat to medium-low; simmer 15 minutes or until buckwheat is tender.
Add bok choy and green onions; cook until tender, about 5 minutes. Stir in vinegar. Ladle soup evenly into 4 bowls. Cut eggs in half lengthwise; place 2 egg halves on each bowl of soup.