Add beef, onion, and water to a large stock pot. Bring to a boil then lower heat and simmer (covered) for 30 minutes. Add radish and continue cooking (covered) about 30 minutes more. At this point, the beef should be tender and shred easily. The radish should also be somewhat soft and translucent, but not fully cooked.
Remove onion and discard. Remove beef and set aside to cool. When the beef is cool enough to handle, shred into bite-sized strips, going along with the grain (not against the grain). Set aside.
Mix spicy sauce ingredients in a bowl and set aside.
In a large bowl, add beef strips, green onions, and Gosari. Add spicy sauce ingredients and mix until everything is well coated.
Add spicy beef/Gosari/green onion mixture to soup and bring to a boil. Lower heat and simmer for 15-20 minutes, to let all the flavors meld. Add bean sprouts and simmer for 5 minutes more.
Taste and adjust seasonings, adding salt and pepper, if needed. Serve hot with rice and kimchi.