Bring the water, kombu, ginger and garlic to a gentle simmer in a medium saucepan over medium heat. Simmer for 30 minutes, then use a slotted spoon to remove the kombu, ginger and garlic.
Add the frozen edamame to the broth and gently simmer over medium heat for 30 minutes. Cool for 10 minutes, then transfer to a blender and purée until smooth (purée in batches if necessary). Return the puréed soup to the saucepan and season with the kosher salt.
Heat a medium skillet over high heat for 1½ minutes. Add the grapeseed oil and mushrooms. Quickly shake the skillet to coat the mushrooms with oil, then let the mushrooms brown, without moving them, for about 2 minutes. Shake the pan and cook until the mushrooms are tender, about 1 minute longer.
Turn the mushrooms out onto a plate and season with the smoked salt. Warm the soup over medium heat, stirring often, then divide among bowls. Squeeze a few drops of lemon juice over each serving, top with smoky mushrooms and serve.