4 pounds duck bones, cut into 2 inch pieces (chicken bones can be substituted)
4-6 quarts cold water
1 (4-by-3-inch) piece kombu
3 large carrots, cut into 1 inch pieces
3 large ribs celery, cut into 1 inch pieces
2 small onions, quartered
2 star anise
3 dried Thai chiles
1 (2-inch) piece ginger (about 1 ounce), peeled and finely sliced
5 large king oyster mushrooms (about 6 ounces total), cut into 1/2-inch dice
1/4 cup soy sauce
3 tablespoons mirin
2 tablespoons fish sauce
Preheat oven to 450°F. Line a baking sheet with parchment paper or foil.
Dry the bones well with paper towels. Place on baking sheet, lightly coat with canola oil, and season with salt and pepper. Roast the bones in the oven until deep golden brown, about 15 minutes. Turn bones over and continue to roast until browned on all sides, about 10 minutes more. Remove from the oven and transfer to a 8-quart or larger stockpot. Add a little water to the baking sheet and use a rubber spatula to scrape all of the fat and browned bits from the tray into the pot.
Add the carrots, celery, onions and garlic to the stockpot with the bones and add enough water to cover by about 2 inches, between 4-6 quarts.
Bring the water up to a boil over high heat and then lower and simmer slowly for 2 to 2½ hours.
While the broth is simmering, rinse the kombu and place in a bowl covered with water. Let soak.
Add to the pot of broth along with the soaked and strained kombu, ginger, star anise and chilies and simmer for another 15 minutes.
Trim the dark green part off the scallions, leaving just the white and light green parts. Heat a large skillet over high heat. Add enough canola oil to coat the bottom of the pan and add the scallions. Sprinkle with a little salt to season. Char the scallions until dark brown and softened, about 1½ minutes. Flip and char the second side, 1½ minutes more. Transfer to a paper towel-lined plate to absorb some of the oil.
Pour a little more oil into pan to coat, set over medium-high heat and add chopped mushrooms. Season with a little salt and ground pepper and sauté until browned and softened, about 3 to 5 minutes.
Add mushrooms to the pot of broth and simmer for 30 minutes more.
Strain and discard solids. Degrease some of the fat off the top of the broth.
Add the soy sauce, mirin, and fish sauce and stir to combine. Taste and adjust seasoning as needed.