Mix the broth base. In a wide, shallow pot or a large deep skillet, stir together the water or stock, kimchi juice, gochujang, gochugaru, soy sauce, sugar, and garlic. This is easier before the pot is full of food.
Lay in the sturdy ingredients. Add the chopped kimchi, onion, Spam, sausages, tofu, mushrooms, and baked beans if using. Arrange them in sections rather than dumping them in a pile, then pour the seasoned broth over everything.
Bring it to a boil. Set the pot over medium-high heat and bring it to a steady boil. Let it bubble for 8 to 10 minutes, until the kimchi softens and the broth turns a deep orange-red. Spoon broth over the tofu and mushrooms as it cooks.
Taste before the noodles. Noodles and cheese will mute the seasoning, so check now. The broth should taste a little too bold at this point: spicy, sour, and salty. Add gochugaru for heat or a pinch of sugar if the kimchi is very sour.
Add the fast ingredients. Nestle the ramen noodles and tteok (if using) into the broth, add the scallions, and cook 3 to 4 minutes, until the noodles are just tender. Keep the noodles submerged and loosen them with chopsticks so they cook evenly.
Finish with cheese. Lay the cheese slices over the center of the pot, put the lid on for 30 seconds, then serve right away. Stir the melted cheese through the noodles at the table.