* - end trimmed, cut into 3/4" thick slices
** - thinly sliced
Lightly brush eggplant slices with oil and put them on a tray.
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). When grill is hot, rub it lightly with a paper towel dipped in salad oil.
Lay eggplant slices close together on grill. 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. Return slices to the tray. If slices are cooked ahead, cover and chill up to a day.
In a bowl, cover cucumber and onion slices with ice water. Quickly squeeze slices to briuse lightly, then cover and chill 30 minutes to 1 hour; drain. In bowl, mix cucumber, onion, vinegar, sugar, and 1/2 teaspoon salt. If made ahead, cover and chill up to 4 hours. Coarsely chop eggplant and spoon onto a rimmed platter; top with cucumber and onion mixture. Add salt to taste.