In a large bowl, add the rice cakes and soak in cold water for 20 minutes.
Remove the brisket from the soup and place in a bowl to cool slightly. Remove and discard the onions and garlic. Return the broth to a boil and season with soup soy sauce, salt, and pepper.
Divide the eggs with the whites in one bowl and the yolks in another. Place a flat skillet over medium low heat and lightly oil. Beat together the egg yolks and add to the pan, tilting to make a thin layer. Once it is just set and not yet browned, flip to cook the other side until just set. Transfer to a plate or cutting board and slice into thin strips.
Using forks, shred the brisket. Toss with the minced garlic, sesame oil, salt, and pepper. Set aside.
Drain the rice cakes and add to the boiling broth along with the dumplings. Stir the bottom of the pot to keep them from sticking. Cook until the dumplings are heated through and the rice cakes are chewy but not completely softened, about 4 minutes. Slowly pour in the egg whites while stirring to create ribbons throughout the soup. Once cooked through, remove from heat.
Divide the soup among serving bowls. Top with shredded beef, sliced eggs, seaweed, and green onions. Serve immediately.