Place the chicken, water, ½ onion, and garlic cloves in a large pot, and bring it to a boil. Skim off any scum that rises to the top as it heats up. Reduce the heat a simmer and let it cook for about 30 minutes.
Add the dashi pack and let it simmer for another 15 minutes or until the chicken breast is cooked through.
Remove the chicken breasts from the pot and let them cool a bit before shredding the chicken meat into bite-sized pieces.
Discard the garlic cloves, onion, and dashi pack from the broth.
Season the broth with the soy sauce and salt, and bring it back to a boil.
Add the prepared vegetables (zucchini, carrots, onion, potatoes), noodles, and some of the shredded chicken and let it simmer until the noodles and vegetables are cooked through (about 3-4 minutes).
Ladle into bowls and serve immediately with a side of fresh kimchi.