Place the chicken breast, celery, carrot, and onion in a large pot. Add enough water to cover the ingredients.
Bring everything to a boil over medium-high heat, then reduce the heat to low and simmer, partially covered, for about 40 minutes.
Turn off the heat and let the chicken cool slightly while still in the liquid.
Once the chicken has cooled for 5-10 minutes, remove the skin and bones. Shred the meat, then set aside. Discard the vegetables.
Heat the sesame oil in a large pot over medium-low heat. Add the scallion whites, ginger, and garlic, then cook, stirring often, until the onions have softened, about a minute or two.
Add the chicken broth, tamari or soy sauce, rice vinegar, and 1 cup of water to the pot.
Increase the heat to medium-high, cover, and bring to a slow boil.
Add the noodles, return to a boil, and cook until the noodles are almost done, about 6 to 7 minutes (or a minute less than the package says).
Stir in the bell peppers, shredded chicken, and spinach. Cook until the spinach wilts, about a minute.
Divide the soup into 4 serving bowls and top with scallion greens.