Put eggs in a small pot of cold water. Bring to a boil, remove from heat, cover, and let sit 15 minutes. Drain; rinse with cold water.
Bring dashi to a boil. Reduce heat to low and add mushrooms, mirin, and soy sauce.
Bring a 3-qt. saucepan of water to a boil. Add soba and cook, stirring to separate noodles, until softened, about 5 minutes. Drain but don't rinse. Divide noodles among 4 serving bowls.
Pour 1 1/2 cups of dashi over noodles in each bowl. Arrange mushrooms and daikon over noodles, dividing evenly so each bowl has a neat row of both. Peel eggs and cut each in half lengthwise, placing 1 half in each bowl. Divide nori and green onions among bowls.
*Find liquid dashi in containers in Japanese groceries and some gourmet stores. It's more widely available as a dry concentrate called dashi-no-moto; reconstitute according to package directions.
Note: Nutritional analysis is per serving.