Grilled Eggplant Salad with Onion and Cucumber

1. * - end trimmed, cut into 3/4" thick slices ** - thinly sliced.
2. 1. Lightly brush eggplant slices with oil and put them on a tray.
3.  2. On a barbecue with lid, place grill 5-6" over a solid bed of hot coals (you can hold your hand at grill level only 2-3 seconds).
4. When grill is hot, rub it lightly with a paper towel dipped in salad oil.  3. Lay eggplant slices close together on grill.
5. Cover barbecue, open drafts, and cook until slices are well browned and soft when pressed, 15-20 minutes; turn, as needed, with a wide spatula.
6. Return slices to the tray.
7. If slices are cooked ahead, cover and chill up to a day.  4. In a bowl, cover cucumber and onion slices with ice water.
8. Quickly squeeze slices to briuse lightly, then cover and chill 30 minutes to 1 hour; drain.
9. In bowl, mix cucumber, onion, vinegar, sugar, and 1/2 teaspoon salt.
10. If made ahead, cover and chill up to 4 hours.
11. Coarsely chop eggplant and spoon onto a rimmed platter; top with cucumber and onion mixture.
12. Add salt to taste.