Bring the anchovy-kelp broth to a boil in a pot with a tight-fitting lid.
Add the bean sprouts and garlic. Cover immediately and do not lift the lid for 5 minutes — letting the steam escape early is the most common reason this soup turns out smelling grassy instead of clean.
After 5 minutes, lift the lid, stir once, and add the salt and guk-ganjang if using.
Simmer uncovered for 3 to 5 more minutes, until the sprouts have softened but still have a slight snap.
Stir in the scallions and gochugaru, if using, and simmer 1 more minute.
Take the pot off the heat, stir in the sesame oil, and top with sesame seeds before serving.