- Arrange eggplant slices in single layer on clean kitchen towels or paper towels.
- Sprinkle both sides liberally with salt.
- Let stand 1 hour to allow bitter juices to drain.
- Preheat broiler or prepare grill.
- Rinse eggplant slices and pat dry.
- Brush slices with 1 tablespoon oil.
- Broil or grill until eggplant slices are soft and slightly charred, 4 to 6 minutes per side.
- Set aside to cool.
- In large skillet, heat remaining 1 tablespoon oil over low heat.
- Add onion and cook 1 minute.
- Add celery and cook, stirring occasionally, until tender, about 10 minutes.
- Add tomatoes (with liquid), and capers.
- Increase heat to medium and cook, stirring occasionally and breaking up tomatoes with spoon, until liquid is slightly reduced, about 7 minutes.
- Meanwhile, coarsely chop eggplant.
- Add to tomato mixture and cook 1 minute.
- Stir in vinegar and sugar and cook, stirring occasionally, until most of liquid evaporates, about 10 minutes.
- Remove from heat and stir in basil.