Bring 4 cups of vegetable broth to a boil. Turn off heat and add dried mushrooms. Cover and allow to soak about 20 minutes. Drain mushroom soaking water into a large pot and add 4 cups of water.
Turn on heat to bring liquid to a boil. As liquid is heating, cut stems off mushrooms and cut mushrooms into very thin slivers.
Cut tofu into very thin matchstick slivers and pour a little soy sauce on top of them. When liquid comes to a boil, reduce heat to medium, add slivered mushrooms and allow to simmer about 10 minutes. Then add slivered tofu and simmer another 5 minutes.
Add next 3 ingredients and, in a little cup, quickly dissolve arrowroot in the cold water. Add to ingredients in pot and stir constantly till the arrowroot thickens. (As the soup thickens, it will become clear again.
It should take only 20 to 30 seconds after liquid comes to a boil for the arrowroot to thoroughly thicken.) Cover and turn off heat. Add sesame oil. The soup should be served immediately, topped with slivered green onions. Makes eight 1 cup servings.