1. Rinse and drain shiitakes. Trim off and discard stems. Thinly slice mushroom caps.
2. In a 4- to 5-quart pan, combine mushrooms, broth, rice, onion, and 2 cups water. Cover and bring to a boil over high heat. Reduce heat and simmer 10 minutes.
3. Meanwhile, scrub clams under cool running water. Discard any that do not close when touched.
4. Add clams to pan, turn heat to high, cover, and bring to a boil again. Reduce heat and simmer until clams open, about 10 minutes more.
5. Stir in mizuna, then ladle soup into wide bowls.