Bring the anchovy-kelp broth to a simmer in a medium pot.
Whisk in the doenjang and gochugaru until they dissolve rather than float as clumps.
Add the beef and napa cabbage. Simmer 15 minutes, until the cabbage has gone fully limp and the beef is tender.
Stir in the garlic and guk-ganjang. Taste and adjust with salt.
Add the bean sprouts and simmer 5 more minutes, until they've lost their raw crunch.
Stir in the scallions. If using egg, drizzle it in a thin stream over the simmering surface and let it sit 20 seconds before a gentle stir.
Finish with a crack of black pepper and serve hot, with rice on the side or spooned directly into the bowl.