In a medium saucepan, heat the olive oil over medium heat. Add garlic and ½ cup of the scallion and cook until softened, about 2 minutes. Mix in the ginger and cook, stirring, until lightly browned, about 1 minute more.
Add broth, shiitake, tamari, and 1 cup water. Bring to a boil over high heat. Reduce heat to medium and simmer until shiitake are just tender, 5 minutes.
Season the chicken with salt and pepper. Stir into the soup. Simmer until chicken is cooked through, about 3 minutes.
Remove the saucepan from the heat. Stir in the spiralized zucchini and sesame oil. Let sit 5 minutes. Season with salt and pepper. Garnish with the remaining scallion, cilantro, sesame seeds, and more sesame oil, if desired.