Rinse the beef cubes in cold water and drain.
In a large pot, add the 10 cups water, beef, garlic cloves, green onion over high heat. Once boiling, reduce to a simmer, cover and cook until the meat is tender, 1-1 1/2 hours. Skim off and discard any foam that develops on the surface.
While the broth is simmering, place the rice cakes in a large bowl and cover with cold water. Allow to soak for 30 minutes and drain.
Place a skillet over medium heat. Place a sheet of the gim on the hot skillet to lightly toast on each side until crisp. Repeat with the other sheet. Slice into strips or crumble with your hands and set aside.
Separate the egg yolks and whites into two separate bowls. Very lightly grease the skillet with oil and place over medium low heat.
Lightly beat the egg yolks until smooth. Once the skillet is thoroughly heated, remove from heat. Immediately add the egg yolks and tilt to create a thin, even layer. Once set on the bottom but not yet browned, about a minute, flip to cook the other side. Remove to a cutting board, thinly slice into strips, and set aside.
Grease the skillet again. Lightly beat the egg whites and pour into the skillet, tilting to form a thin, even layer. Once set, but not browned, flip to cook the other side. Remove to a cutting board, thinly slice into strips, and set aside.
Once the beef is tender, strain the broth and place back over medium heat. Shred the beef pieces and place in a small bowl. Discard the vegetables. Season the shredded beef with the minced garlic, 1 teaspoon sesame oil, and black pepper. Set aside.
Season the broth with the soup soy sauce and sesame oil. Add salt and freshly ground black pepper to taste. Once boiling, add the drained rice cakes and cook, stirring occasionally, until softened, 5-8 minutes.
Divide the soup among individual bowls. Top with the seasoned beef, crumbled seaweed, egg strips, and reserved green onion. Serve immediately.