In a large Dutch oven, heat canola oil over medium heat. Add onion, carrot, ginger, garlic, and a heavy pinch of salt. Cook for 7 minutes, stirring occasionally, or until veggies begin to take on some color.
Add shiitake stems, kombu, 1/2 ounce dried shiitake mushrooms, bok choy, and green onions. Add vegetable stock and 4 cups of water. Bring to a boil over high heat then reduce to a simmer.
Cover and cook 25 minutes while your roast your vegetables.
After broth has simmered 25 minutes, strain broth through a fine-mesh strainer or chinois into a large clean bowl. With the back of a wooden spoon, press the liquid out of the stock veggies, mashing them with the spoon. Some mashed veggies will collect on the bottom of the strainer, scrap those from the strainer into the bowl and discard the rest of the veggies.
Return broth to the Dutch oven and season to taste with salt, keeping in mind soy sauce will be added later.