Heat sesame oil in a large pot or deep skillet over medium heat. Add sliced onion and garlic, sauté until fragrant and onions become translucent.
Add sliced carrots, shiitake mushrooms, vegan hot dogs, and vegan kimchi. Cook for about 5 minutes until the vegetables begin to soften.
Pour in vegetable broth (or water) and bring the mixture to a simmer.
Add gochujang, gochugaru, and soy sauce (or tamari). Stir well to combine all the flavors.
Add firm tofu cubes and canned baked beans or chickpeas if using. Stir gently to incorporate.
Toss in the sliced cabbage or Napa cabbage. Let it simmer for about 10-15 minutes until the vegetables are tender and the flavors have melded.
Meanwhile, cook the instant ramen noodles separately according to the package instructions. Drain and set aside.
Taste the stew and adjust seasoning with salt and pepper as needed.
To serve, place a portion of cooked ramen noodles into individual serving bowls. Ladle the hot stew over the noodles.
Garnish with chopped green onions and serve hot.