In a large pot, add the stock, garlic, and ginger. Cover the pot with a lid and heat over medium-high heat until it comes to a boil.
Add the ground beef and green onions and cook for about 5 minutes while breaking up the chunks of beef using a wooden spoon or spatula.
Add the rice cakes and cook according to the packaging. Mine took about 5 minutes until the rice cakes floated to the top, but others may take a little longer.
Turn the heat off and add soy sauce and sesame oil. Taste for seasoning and add salt and pepper to taste. If adding kimchi, add it now. Pour into bowls and garnish with seaweed and sesame seeds.
Notes
Serve immediately. The rice cakes can get mushy the longer it gets left in the soup, so serve this soup as soon as it's ready.
Enjoy leftovers! If you plan to have leftovers the next day, don't add all of the tteok, as they don't taste as good the following day. Instead, only cook half the amount of tteok, finish it in one sitting, and add the other half of tteok the next day when you're reheating the soup on the stove.