Score the cut side of each eggplant half in a shallow crosshatch pattern, being careful not to cut through the skin.
Preheat the broiler. Brush the cut sides of the eggplant generously with oil and place cut-side up on a baking sheet.
Broil for 8-10 minutes until the eggplant is tender and lightly browned.
While the eggplant broils, whisk together the miso paste, mirin, sake, and sugar in a small bowl until smooth.
Remove the eggplant from the broiler and spread the miso glaze evenly over the cut sides.
Return to the broiler for 2-3 minutes, watching closely, until the glaze is bubbling and lightly caramelized in spots.
Sprinkle with toasted sesame seeds and sliced scallion. Serve hot over rice.