In a large pot, add the chicken, onion, garlic, ginger, and chicken broth. Bring to a boil over medium heat, then reduce the heat to low and simmer for about 1 hour, or until the chicken is tender and cooked through.
While the chicken is simmering, prepare the noodles. In a mixing bowl, combine the all-purpose flour and salt. Gradually add water, a little at a time, and mix until a soft dough forms. Knead the dough for about 5 minutes, then cover it with a damp cloth and let it rest for 30 minutes.
After the dough has rested, roll it out on a floured surface until it is about 1/8-inch thick. Cut the dough into thin strips, about 1/4-inch wide.
Bring a pot of salted water to a boil. Add the noodles and cook for about 5 minutes, or until they are al dente. Drain the noodles and rinse them under cold water to stop the cooking process.
Once the chicken is cooked, remove it from the pot and shred it into bite-sized pieces, discarding the bones and skin. Return the shredded chicken to the pot and season with salt and pepper to taste.
In a separate pan, heat the vegetable oil over medium heat. Add the sliced green onions and sauté for a few minutes until they are fragrant.
Add the sautéed green onions to the pot of chicken and broth. Stir to combine all the ingredients, and let the soup simmer for an additional 10 minutes to allow the flavors to meld together.
To serve, divide the cooked noodles among bowls, then ladle the hot soup over the noodles. Garnish with additional sliced green onions, if desired.
Enjoy your homemade Dak Kalguksu Soup while it's hot!