If using bacon or sausage, cook it first in the pot until it renders some fat and browns slightly, then set aside, leaving the fat behind.
Add the broth, gochujang, gochugaru, soy sauce, sugar and garlic to the pot. Whisk until smooth.
Add the rice cakes. Simmer over medium heat 6 to 8 minutes, stirring occasionally, until they've softened and the sauce has reduced slightly.
Add the fish cake (or return the cooked bacon/sausage to the pot). Simmer 3 more minutes.
Lower the heat before adding the cream — this matters more than it sounds like it should.
Stir in the cream and let it warm through for 1 to 2 minutes without letting it come to a hard boil. It should turn from red to a rounded pink-orange, not separate or look grainy.
Scatter the mozzarella on top if using and let it melt in the residual heat, off the burner. Top with scallion and serve.