Soak the rice cakes in cold water for 20 minutes if refrigerated or frozen; skip this if they're fresh and soft.
Whisk the gochujang, gochugaru, soy sauce, sugar and garlic into the hot broth until smooth.
Add the drained rice cakes and simmer over medium heat for 7 minutes, stirring every minute so they don't stick.
Add the fish cake and simmer 3 more minutes, until the sauce has visibly thickened and clings to a lifted rice cake.
Break the ramyeon noodle brick roughly in half and add it to the pan along with about a third of its seasoning packet, to taste.
Cook the noodles for exactly 2 minutes at a hard simmer, no longer, then take the pan off the heat immediately — they keep softening in the residual heat.
Stir in the scallions, lay the cheese slice over the top if using, and crack the egg in whole, letting the retained heat set the white while the yolk stays soft.
Serve straight from the pan within a few minutes, while the noodles still have some bite left.