Cut the ends off Shimeji mushrooms and pull them apart. Slice carrot into thin quarter rounds and thinly slice green onion. Set aside.
Make dumplings. Peel and grate potatoes, add flour, and mix well. Season with salt. Adjust the amount of flour to achieve a batter that is neither too loose nor too hard.
Bring the Dashi to a boil, then add Shimeji mushrooms and carrots, cooking for 2 minutes. Drop spoonfuls of the potato mixture into the soup one by one. Cook for 4-5 minutes or until the dumplings are set. Do not stir while the dumplings are setting.
Dissolve miso paste in the soup and add sliced onion. Serve immediately.