Soak scallion in ice water until crisp, at least 10 minutes. Drain.
Meanwhile, bring mirin, sake, soy sauce, and zarame sugar to a boil in a small saucepan; reduce heat and simmer until slightly reduced, 12–15 minutes. Set tare aside.
Prepare grill for medium heat. Thread 3 mushroom pieces onto each skewer. Brush lightly with oil and season lightly with salt. Grill until golden brown, about 1 minute per side. Then cook, brushing with tare and turning occasionally, until glazed and tender, about 2 minutes more. Serve mushrooms topped with scallions.
