In a large pot, heat the sesame oil over medium heat. Add the grated ginger and minced garlic and sauté for 1 minute.
Add the sliced zucchini to the pot and cook for 2-3 minutes until slightly softened.
Pour in the chicken broth and bring to a simmer. Cook for 10 minutes.
In a small bowl, whisk together the miso paste and soy sauce until smooth.
Add the shredded chicken to the pot and then stir in the miso mixture. Cook for an additional 5 minutes.
Season with salt and pepper to taste.
Serve the soup hot, garnished with chopped green onions.