Rinse the beef brisket cubes under cold water to remove impurities; drain well and set aside.

In a large pot, combine 8 cups of filtered water, the rinsed beef, whole garlic cloves, and both parts of the chopped green onions. Bring to a vigorous boil over medium-high heat. Once boiling, reduce to low, cover, and let simmer for 1 to 1.5 hours.

Soak the sliced rice cakes in cold water for about 30 minutes. After soaking, drain the rice cakes and set aside.

Toast the sheets of gim in a dry skillet over medium heat for 1-2 minutes until crispy. Crumble or slice them into small pieces for garnish.

Heat a non-stick skillet over medium heat. Separate the eggs, pour the yolks into the skillet and cook until just set. Slice into strips and repeat with the egg whites.

After the beef is tender, strain the broth back into the pot. Shred the beef and set aside, discarding the strained vegetables.

Add the Korean soup soy sauce and sesame oil to the strained broth; season with salt and black pepper to taste. Bring to a gentle boil.

Add the soaked rice cakes to the boiling broth and cook for 5 to 8 minutes, stirring gently until they become tender and chewy.

Ladle the hot broth and rice cakes into bowls, topping with shredded beef, egg strips, crumbled gim, and any remaining chopped green onions.