Once the chicken is completely cooked through, turn the heat off under the pot. Remove the chicken to a large plate or cutting board to cool a little. You can do this with 2 very large spoons or spatulas, or tongs. Fish out the garlic, ginger, and onion from the pot and discard, leaving the chicken stock in the pot.
Strain the stock. Remove rendered fat by carefully skimming it off the surface with a large spoon, using a fat separator, or turbo-cooling the stock in an ice bath then refrigerating the stock overnight for the fat to harden. (This makes the process a 2 day affair!)
When the chicken is cool enough to handle, remove and discard the skin. Pull the meat off the bones, shredding into thin strips.
Combine shredded chicken, 4-5 stalks green onions, and all the broth with the vegetables and seasonings in whichever pot is bigger. Bring to a boil, then turn down heat and simmer until green onions are softened, about 10 minutes.
Add 4-6 ounces cooked Korean sweet potato glass noodles aka "dang-myun" if using. Add the lightly beaten 4 large eggs to the soup pot and continue simmering without stirring until eggs are cooked through, about 8 minutes. Drizzle in 1 tablespoon sesame oil just before serving.
Ladle soup into individual serving bowl. Drizzle with extra sesame oil for added umami.