1/2 ounce (about 1/2 cup) dried shiitake or mixed dried mushrooms
3 cups boiling water
1 tablespoon peanut oil or canola oil
2 cups diced onion
3 cloves garlic, thinly sliced
6 1/8-inch-thick slices peeled fresh ginger
6 cups reduced-sodium chicken broth
1/4 cup reduced-sodium soy sauce
1 2-to-3-inch stick cinnamon
1 whole star anise
1 teaspoon freshly ground pepper
2 pounds boneless, skinless chicken thighs, trimmed and cut into 1-inch pieces
1 bulb fennel, cored and cut into 1-inch pieces
8 scallions, whites cut into 2-inch pieces and greens chopped, divided
1 pound bok choy, preferably baby bok choy, white stems sliced lengthwise and greens chopped, divided
2 cups (4 ounces) mung beans sprouts
1/2 cup chopped fresh cilantro
2 teaspoons toasted sesame oil
Lime wedges for garnish
Place mushrooms in a heatproof measuring cup and cover with boiling water. Soak for at least 30 minutes or up to several hours. Remove the mushrooms from the water, remove and discard stems (if any) and cut into 1/8-inch slices; set aside. Strain the soaking liquid and reserve.
Heat oil in a large soup pot or Dutch oven over medium heat. Add onion, garlic and ginger and cook, stirring, for 5 minutes. Pour in the reserved mushroom liquid, broth, soy sauce, cinnamon stick, star anise and pepper. Bring to a boil. Reduce to a simmer and stir in chicken. Simmer for 20 minutes.
Stir in fennel, scallion whites and the reserved mushrooms and cook for 5 minutes. Add bok choy stems, return to a simmer and cook for 3 minutes more. Stir in bok choy greens and bean sprouts. Cook until the greens are just wilted, about 2 minutes more.
Discard the cinnamon stick and star anise. Ladle the soup into bowls. Garnish each bowl with scallion greens, cilantro and a 1/4-teaspoon drizzle of sesame oil. Serve with lime wedges, if desired.