Peel the carrot and cut into rounds. If desired, use a flower cutter to shape the rounds. Set aside.
Remove the stems from the shiitake mushrooms and cut slits across the tops if desired. Set aside.
Place the kirimochi on a baking sheet lined with aluminum foil about 2 inches (5 centimeters) apart. Place in the oven under a low broil until toasted and inflated.
Return strained dashi (water cooked with kombu and bonito) to cleaned large pot over high heat. Stir in the sake, soy sauce, and salt. Once boiling, reduce heat to a simmer and add the marinated chicken pieces.
Cook the chicken until tender, about 5-10 minutes, skimming off any foam that forms at the top of the soup.
In another large pot, bring water to a boil over high heat. Place a bowl of ice water near the pot. Add the Komatsuna or spinach to the boiling water and cook just until blanched and tender, about 30 seconds to 1 minute.
Drain the Komatsuna and immediately transfer to a bowl of ice water. Once cool enough to handle, squeeze out any excess water and cut the Komatsuna into pieces about 2 inches (5 centimeters) long.
Add the cut carrots and mushrooms to the soup. Continue to simmer until tender, another 5 minutes.
Add the sliced Kamaboko and cook just until heated through.
Divide the cooked mochi among 4 soup bowls. Top with pieces of the chicken thighs, then the Komatsuna.
Pour in the soup and garnish with the carrot slices, mushrooms, and Kamaboko. Top the Komatsuna with a few pieces of yuzu zest.
Add 1 sprig of Mitsuba to each bowl and serve immediately.