Prep the kimchi and pork: cut the kimchi into bite-size pieces over a bowl to catch the brine, measuring about 2 cups chopped kimchi and 1/2 cup brine. Cut the pork into pieces about the same size as the kimchi.
Start the pork and aromatics: heat the oil in a heavy pot over medium heat. Add the pork and cook 3 to 4 minutes until the edges lose their raw color and some fat renders. Add the onion and cook 2 minutes until slightly softened. Stir in the garlic and ginger and cook 30 seconds until fragrant.
Fry the kimchi briefly: add the chopped kimchi and cook, stirring often, for 4 to 5 minutes until the cabbage darkens slightly and the edges look translucent.
Build the broth: pour in the kimchi brine and stock. Stir in the gochugaru, soy sauce, and sugar. Bring to a boil, then lower to a steady simmer.
Simmer until tender: simmer, partially covered, for 15 to 18 minutes until the kimchi is soft and silky but still holds its shape. Taste and adjust with more soy sauce, sugar, or brine as needed.
Add tofu and scallions: lay the tofu slices on top and spoon broth over them. Scatter the scallions over the top. Simmer 3 to 4 minutes until the tofu is heated through. Do not stir hard or the tofu will break.
Finish and serve: drizzle the sesame oil over the top and serve bubbling hot with steamed rice.