Sprinkle the eggplant with the coarse salt and toss the cubes. Let them drain in a colander for 30-60 minutes, then pat them dry with paper towels. As it sits, you will see a dark liquid emitted and that is the "bitterness" of the eggplant.
Preheat oven to 375 degrees.
Heat 1/4 cup olive oil in a 12" skillet and saute the onion, peppers and celery over moderately high heat for 5 minutes, stirring as it cooks.
Add the remaining 1/4 cup of olive oil and let it heat. Once it's hot, add the eggplant and cook for another 5 minutes. Add the tomato puree, red wine vinegar, sugar and garlic and cook for another 2 minutes.
In the skillet (if the handle is oven-proof) or transfer to a baking dish, bake the caponata uncovered for 20 minutes, stirring occasionally. Add the olives and capers and stir well. Bake another 30 minutes or a little longer until most of the liquid has evaporated, stirring frequently to get it to the right consistency which is thick not liquidy.
Cool the caponata, then taste and season with salt and pepper. Refrigerate in a covered dish for 24 hours before serving. The caponata will keep for a week in the refrigerator or at least six months if placed in the freezer in plastic containers.