Soak in cold water for 10 to 20 minutes if hardened.
In a large pot, bring the meat, onion, scallions and garlic to a boil in 14 cups of water. Reduce the heat to medium low, and skim off the scum. Simmer, covered, until the meat is tender enough for shredding, 1 to 1-1/2 hours. Remove the meat and cool. Discard the vegetables. Stir in soup soy sauce, salt and pepper to taste.
Cut the cooled beef into 1 – 1 1/2-inch wide strips, shred, and combine well with garlic, sesame oil, and salt and pepper to taste.
To make egg garnish (jidan), separate the egg white and yolk. Lightly beat the white by gently cutting it with a spoon. Stir the yolk with a spoon until smooth. Heat a lightly oiled nonstick skillet over medium low heat. Pour each egg part into a thin layer, tilting the skillet and/or spreading with a spoon. Cook each side briefly. (Do not brown the egg.)
Roll each egg crepe, and slice into short thin strips. Slice the scallion diagonally into thin strips. Roast the gim on a hot skillet. Cut into thin 1 1/2-inch strips with kitchen shears, or simply crush them with hands.
Return the broth to a boil. Add the rice cake slices and boil until soft, usually about 5 – 8 minutes. Ladle the steaming soup into individual bowls and garnish with the shredded beef, egg, scallion and gim strips.
Although egg jidan is a classic garnish for tteokguk, a common alternative is to drizzle a lightly beaten egg over the boiling soup right before turning the heat off. Or you can omit the egg part entirely if you like.
If you don't like your soup too thick and starchy, you can parboil the rice cake slices in the boiling water until soft, drain, and then add to the soup.
If you want to add some mandu (dumplings), you can drop them in along with the rice cake slices, stirring gently so they don’t stick to the bottom of the pot. Boil until all of them float, and continue to cook for another minute or two. You can also steam mandu or cook in boiling water separately, and add to the soup when the rice cake is softened. This method will keep the soup from becoming too thick from all the starch released from the rice cake and mandu.