In a large pot, combine the quartered chicken, 12 cups of water, halved onion, garlic cloves, and sliced ginger. Bring to a boil over high heat.
Once boiling, reduce heat to medium-low and simmer uncovered for 30 minutes, skimming off any foam or impurities that rise to the surface.
Remove the chicken pieces from the broth and set aside to cool slightly. Strain the broth through a fine mesh sieve into another pot, discarding solids. Return the broth to the stove and keep warm over low heat.
When the chicken is cool enough to handle, remove the meat from the bones and shred it into bite-sized pieces. Discard the bones and skin.
To make the noodles, combine the all-purpose flour and 1/2 teaspoon salt in a mixing bowl. Gradually add 3/4 cup water while stirring until a dough forms.
Transfer the dough to a floured surface and knead for about 5 minutes until smooth and elastic. Cover with a damp cloth and let rest for 20 minutes.
After resting, roll the dough out to about 1/8-inch thickness. Cut into thin strips about 1/4 inch wide to form noodles.
Bring a separate pot of water to a boil. Add the noodles and cook for 3 to 4 minutes or until tender but still chewy. Drain and rinse under cold water to stop cooking. Set aside.
Return the strained broth to medium heat. Add the soy sauce, salt, and black pepper to taste.
Add the julienned zucchini and carrots to the broth and cook for 3 to 4 minutes until tender but still crisp.
Add the shredded chicken and cooked noodles to the broth. Stir gently to combine and heat through for 2 minutes.
Turn off the heat and stir in the sesame oil.
Ladle the soup into bowls and garnish with sliced green onions and toasted sesame seeds.