Heat a grill or grill pan over medium-high, or heat the oven to 425F (220C) with a foil-lined baking sheet.
Trim the stems and halve each eggplant lengthwise. Score the cut side in a 1/4-inch-deep crosshatch without piercing the skin. Brush the cut faces generously with the neutral oil.
Grill cut-side down 5-7 minutes until deeply browned, flip and cook skin-side down 4-5 minutes more until a fork slides into the flesh with no resistance. (Oven: roast cut-side up 18-20 minutes until completely soft.)
Meanwhile, stir the miso, sugar, mirin, sake and sesame oil in a small saucepan over low heat 2-3 minutes until the sugar dissolves and the glaze is smooth and glossy. Do not boil hard.
Heat the broiler to high with a rack about 6 inches from the element. Arrange the eggplant cut-side up on the baking sheet and spread the glaze thickly over each half, about 1 tablespoon per half. Broil 3-5 minutes, watching closely, until the glaze bubbles, caramelizes and chars in spots.
Sprinkle with toasted sesame seeds and scallions. Serve hot or warm.