Shiitake mushrooms can be used in place of the matsutake, which are sometimes difficult to find.
Gently clean the mushrooms with a mushroom brush or paper towel. Trim tough stem ends, and discard. Cut the mushrooms lengthwise into 1/4-inch-thick slices, keeping the cap and stem attached.
In a medium bowl, whisk together the soy sauce, mirin, and ginger. Add the mushrooms, and let stand at room temperature 30 minutes.
Heat grill to medium-hot. Grill mushrooms in batches until they are golden and start to curl, 2 minutes per side.
In a small saucepan, bring soy mixture to a simmer over medium heat; cook until reduced and almost syrupy, 5 minutes. Remove from heat, and toss with grilled mushrooms. Transfer to a serving bowl. Sprinkle with scallions just before serving.