Score the cut side of each eggplant half in a crosshatch pattern. Salt and let sit for 20 minutes.
Pat the eggplant dry. Preheat broiler to high with rack 6 inches from heat.
Brush cut sides with oil and broil for 8-10 minutes until tender and lightly golden.
Whisk together miso, brown sugar, rice vinegar, and water until smooth.
Spread the miso glaze over the eggplant. Broil for 4-5 minutes until caramelized and bubbling.
Toss warm rice with sesame oil, sesame seeds, and a pinch of salt.
Spoon sesame rice into bowls, top with eggplant, and finish with sesame seeds and green onion.