In a heatproof glass bowl, combine the water and sugar and microwave at high power for 10 seconds, until the sugar is dissolved. Add the grated ginger, vinegar, sesame oil and the 1/2 cup of vegetable oil and season with salt.
Set a large steamer basket in a large saucepan, add 1/2 inch of water and bring to a boil over moderately high heat. Steam the eggplant in batches, cut side down, until just tender, about 4 minutes.
Heat a grill pan. Lightly brush the eggplant all over with vegetable oil. Grill over high heat until grill marks appear, about 30 seconds per side, then transfer to a serving platter. Spoon the ginger sauce over the eggplant. Garnish with the purple basil (if using) and serve at once.