Heat olive oil and sesame oil over medium heat until it begins to lightly smoke. Lower the heat, add the white parts of the green onions, and sauté until softened but not browned, about 3-5 minutes.
Add the carrots and celery and sauté them for 3-5 minutes. Add the chicken broth and increase the heat to a boil.
Place the green tea, garlic, ginger, thyme, and bay leaf in a 12 inch by 12 inch square of cheesecloth, tied into a small satchel with some butchers twine (this is called a bouquet garni.)
When the broth reaches a full boil, lower the heat to a slow simmer, then remove the green tea bouquet garni after a minute. Add the miso paste, cover the pot, and let the soup simmer for 15 minutes on low.
After 15 minutes on low heat, add the noodles, the shredded chicken, and mushrooms, then bring the soup back to a boil. Serve piping hot with sliced green onion and cilantro, then add hot sauce to taste and serve.