Put the flank steak in a pot with the water, onion, 6 whole garlic cloves, and ginger. Bring to a boil, then drop to a low simmer. Skim off any gray foam that rises in the first 10 minutes. Cover partway and simmer 50 to 60 minutes, until the beef pulls apart easily with a fork.
Lift out the beef and set it on a board to cool. Strain the broth and discard the onion, ginger, and boiled garlic. You want about 6 cups of broth left, so top up with water if it reduced too far. Skim the fat off the top if you like a cleaner soup.
When the beef is cool enough to handle, shred it along the grain into rough strips with your fingers or two forks.
In a small bowl, make the chili base: stir together the gochugaru, sesame oil, soy sauce, the 2 minced garlic cloves, and the gochujang if using. It should look like a loose red paste.
Return the broth to the pot over medium heat. Stir in the chili base until the broth turns deep red. Add the shredded beef, mushrooms, and gosari if using. Simmer 15 minutes so the flavors marry.
Add the mung bean sprouts and scallions and simmer 5 minutes more, until the sprouts are tender but still have a little bite. Taste and add salt as needed.
Bring the soup to a gentle boil. Drizzle the beaten eggs in a slow stream around the pot, wait 10 seconds, then stir once so they set in soft ribbons. Serve hot, with steamed rice on the side.