Heat 1 tablespoon avocado oil or other neutral cooking oil in a pot over medium high heat. Add ½ yellow or white onion and 2 cloves garlic and cook until onions begin to turn translucent and garlic becomes fragrant, about 3 minutes.
Add 2 tablespoons gochugaru, 1 tablespoon soy sauce, and 1 teaspoon fish sauce if using, and stir to combine and lightly toast the gochugaru, about 1 minute.
Pour in 2 cups broth or stock, turn up heat and bring to a boil.
Once the stock comes to a boil, add 2 ounces shiitake mushrooms, 2 ounces king oyster mushrooms, 1 medium zucchini sliced, and silken tofu. Bring back to a boil, then turn down heat to medium and simmer until mushrooms are cooked through, about 4 minutes. Taste soup base and season with coarse sea salt (or more soy sauce) as needed.
Just before serving, crack whole 2 eggs into pot, and do not disturb; you are essentially poaching eggs in the soup base. Boil until your preferred cook temperature for eggs, about 3 minutes for set white and runny egg yolk, up to 5½ minutes for a solid yolk (my preference).
Using a large, deep ladle, spoon the Soon Doo-boo Jji-gae into serving bowls, evenly distributing the mushrooms, vegetables, and tofu, and including one whole egg for each serving. Garnish each bowl with a drizzle of toasted sesame oil, sliced green onions, and 2-4 slices jalapeño pepper and fresh cracked black pepper if using.