Prepare your sundubu or soft tofu. If using sundubu in a tube, slice the tube in half (or follow the dotted lines), and slice the tofu into a few pieces. If using a block of sudubu, carefully remove it from the tray of water and slice into a few pieces. Note that you can opt not to slice the tofu and simple break them into smaller pieces when placing in the stew later on.
Heat a medium sized pot or a over medium. Once hot, add in the sesame oil. Sauté the onion and leeks. Sauté for 2-3 minutes until translucent and cooked through. Add in the garlic, gochugaru, and soy sauce. Cook for 2 minutes.
Add in the king oyster or shiitake mushrooms and sauté for a few minutes until lightly cooked through, around 3 minutes. Add in the zucchini and mix well. Pour in the vegetable broth or water. Cover the pot and leave to boil over medium high heat, around 7 minutes.
Once the zucchini and mushrooms are cooked through, taste soup and season with salt as needed. Add in the tofu and leave to boil for 2-3 more minutes. Enjoy while hot with some steamed rice and your choice of sides.