In a medium saucepan, combine the mushrooms with 3 cups cold water. Bring to a boil over medium heat. Remove from the heat and use a fork or tongs to transfer the mushrooms to a cutting board. As soon as the mushrooms are cool enough to handle, trim off and discard the stems and slice the caps very thinly. Return the sliced mushrooms to the pan of water. Bring back to a simmer and then reduce the heat to low. Let the mushrooms steep for 15 minutes (the water needn't be simmering, but it's fine if it does). Taste one of the mushroom slices. If you like the texture, leave the mushrooms in; otherwise, fish them out and discard them.

Bring the mushroom broth to a simmer over medium heat. Add the spinach and tofu and simmer for 1 minute. Remove from the heat. In a small bowl, combine the miso with 2 tablespoons of the broth and mix well. Stir the thinned miso into the soup. Sprinkle each serving with 1 tablespoon of the scallion greens and 1/4 teaspoon of the sesame oil.