Into a large stock pot over high heat, add the sesame oil and allow it to heat until it just starts to smoke. Add the green onions, daikon, garlic, ginger, and star anise. Allow the ingredients to lightly blacken, stirring occasionally; this should take about three minutes; this process of lightly burning the aromatics adds a ton of flavor to the broth.
Add the water to the pot. Using a wooden spoon, scrape up any brown bits from the bottom. Add the carrots, kombu, dried mushrooms, soy sauce, and vinegar to the pot and bring to a boil. Once the water starts to boil, reduce the heat, cover, and simmer for 40 minutes.
Using a spider tool or small strainer, fish out the daikon and mushrooms. Set aside to cool. Using the spider or strainer, remove and discard all other solids from the broth. Crank the heat back up to high and continue to cook the broth until it has reduced by about 20 percent. Makes 4 quarts.