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Cook mushrooms and 1 teaspoon salt in a large, dry saucepan over medium heat, stirring occasionally, until mushrooms collapse, darken slightly, and most of moisture cooks out, 12 to 15 minutes.
Add oil and chopped scallions; cook until scallions are tender, 2 to 3 minutes. Stir in ginger, garlic, and miso and cook just until fragrant, about 30 seconds. Add broth; bring to a boil. Reduce heat to low; simmer 5 minutes.
Reserve 2 tablespoons coconut milk, then add the rest to a blender with mushroom mixture (work in two batches, if needed). Purée until very smooth, about 2 minutes, removing blender cap and covering hole with a folded kitchen towel to allow steam to escape.
Return mixture to saucepan and bring to a simmer over medium heat. If a thinner soup is desired, stir in more broth, a little at a time. Season with salt and white pepper and serve, drizzled with remaining coconut milk and topped with fried mushrooms and scallion tops.
12 ounces cremini or button mushrooms, thinly sliced, plus more for serving (optional, see cook's note)
Kosher salt and freshly ground white pepper
2 teaspoons vegetable oil
4 scallions, white and light-green parts chopped (½ cup), tops thinly sliced on the bias for serving
4 teaspoons minced fresh ginger (from a 1 ½-inch piece)
1 tablespoon minced garlic
1 tablespoon white or yellow miso
2 ½ cups vegetable broth, plus more as needed
1 can (13.5 ounces) coconut milk