In a large pot over medium heat, heat oil. Add onions, garlic and ginger and cook until aromatic. Add chicken and cook, stirring regularly, until lightly browned and juices run clear. Add kinchay and cook, stirring regularly, for about 2 to 3 minutes or until limp. Add fish sauce and cook for about 2 to 3 minutes.
Add chicken broth and bring to a boil, skimming any scum that floats on top. Lower heat, cover and continue to cook until chicken is cooked through.
In a bowl, combine about 1/2 cup of the hot broth and atsuete powder. Stir until atsuete is dissolved. Add atsuete water to pot.
Add carrots and cook for about 1 to 2 minutes or until half done. Season with salt and pepper to taste. Add sotanghon noodles and push down into broth to soften, stirring gently to separate. When noodles have slightly softened, add cabbage and cook for another 2 to 3 minutes or until noodles are cooked and vegetables are tender yet crisp.
Ladle soup into bowls and top with boiled eggs, fried garlic bits and green onions. Serve hot.