Simmer the brisket: Place the beef brisket in a large pot with water, onion, and half the garlic. Bring to a boil, then skim off the gray foam that rises for the first 10 minutes.

Lower the heat: Reduce to a gentle simmer where the surface barely trembles. Cook for 60 to 90 minutes, until a fork slides into the meat with no resistance.

Shred the beef: Remove the brisket and let it cool slightly, then shred it against the grain into thin strands while still warm. Set the shredded beef aside and strain the broth to remove the onion and garlic.

Make the spice paste: In a bowl, mix gochugaru, remaining garlic, gochujang, soy sauce, and 2 tablespoons sesame oil into a thick, deep red paste.

Prepare the fernbrake: If using dried fernbrake, drain it well and cut into 2-inch pieces. If substituting shiitake, slice the soaked mushrooms into thin strips.

Return beef to broth: Pour the strained broth back into the pot and add the shredded beef. Bring back to a simmer over medium heat.

Add the spice paste: Stir the spice paste into the simmering broth until it dissolves and the liquid turns a rich orange-red color. Let it simmer for 5 minutes to cook off the raw chile smell.

Add vegetables: Stir in the scallions, bean sprouts, and fernbrake (or shiitake). Simmer for 15 to 20 minutes, until the scallions soften and the sprouts turn translucent at the edges.

Season the soup: Taste the broth and add salt as needed, along with the remaining tablespoon of sesame oil for aroma.

Add the eggs: Drizzle the beaten eggs into the simmering soup in a thin stream, stirring once gently, so they set into loose ribbons instead of clumps.

Rest and serve: Let the soup sit off the heat for 2 minutes so the flavors settle, then ladle into bowls alongside steamed rice.