Make the Broth:

1. Simmer the broth. Add the water, kombu, dried shiitake mushrooms, halved onion, and smashed garlic to a medium pot. Bring to a gentle simmer over medium heat.
2. Cook low and slow. Once simmering, reduce the heat to low and let simmer for 20-25 minutes. The broth should develop a deep, savory flavor and turn slightly amber.
3. Strain and season. Remove and discard the kombu, onion, and garlic. Remove the shiitake mushrooms and set aside. You can slice them and add them back to the soup if desired. Stir in 2 tablespoons of soy sauce. Taste the broth. It should be rich, savory, and slightly salty. Set aside.

Build the Soup Base:

1. Heat the oil. In a Korean earthenware pot (ttukbaegi/dolsot), a medium-sized pot, or a deep skillet, heat the neutral oil over medium heat.
2. Cook the aromatics. Add the sliced onion and serrano peppers (if using) and cook for 3-4 minutes until softened and slightly translucent. Add the minced garlic and cook for 1 minute until fragrant.
3. Toast the spices. Add the gochugaru (start with 2 tablespoons for medium spice) and gochujang. Stir everything together and cook for 1-2 minutes, stirring constantly. The oil should turn a deep red color and smell incredibly aromatic.
4. Add seasonings. Add 1 tablespoon of soy sauce, sugar, and sesame oil. Stir to combine.

Finish the Stew:

1. Add the broth. Pour in the prepared broth (about 3-4 cups, depending on your pot size and desired consistency). Bring to a simmer.
2. Add vegetables. Add the zucchini and, if using, sliced shiitake mushrooms, and simmer for 3-4 minutes, until tender.
3. Add the tofu. Gently add the entire block of silken tofu to the pot. Use a spoon to break it into large, irregular chunks. Traditionally, people break tofu by hand or with a spoon rather than pre-cutting it into neat cubes.
4. Finish with green onions. Add the green onion pieces. Simmer for 3-5 minutes, letting the tofu heat through and the flavors meld.
5. Taste and adjust. Taste and adjust the seasoning with salt or more soy sauce, as needed. The soup should be deeply savory, slightly spicy, and well-balanced.

Serve:

1. Serve bubbling hot. Carefully transfer the earthenware pot to a trivet or heat-safe surface (if using ttukbaegi, it will be extremely hot). Otherwise, ladle the soup into individual bowls.
2. Garnish and enjoy. Garnish with sesame seeds and extra chopped green onions if desired. Serve immediately with a bowl of steamed white rice and kimchi on the side. The traditional way to eat this is to add spoonfuls of rice directly into the bubbling stew.