1) To make the bread: Mix and knead all of the ingredients to make a soft, supple dough.
2) Place the dough in a lightly greased container, cover with lightly greased plastic wrap, and let it rise until it's very puffy, 45 minutes to 1 hour.
3) Gently deflate the dough, and divide it into four pieces.
4) Grease four 10" squares (or 9" x 12" rectangles) of parchment paper or waxed paper.
5) Use your greased fingers to press each piece of dough onto the parchment, pressing it into a circle or oval. The dough should be no more than 1/4" thick.
6) Brush or spray the crusts with olive oil, and let them rest for about 30 to 60 minutes, while you preheat your grill to about 400°F.
7) Place the crusts with their parchment directly onto the grill, with the parchment on top. Peel off the parchment. If your grill has a cover, close it.
8) Cook the breads for about 4 minutes. Turn one over; it should feel fairly set, and have golden-brown sear marks. If not, continue cooking till the breads feel set and are golden on one side.
9) Cook the breads on the other side for about 4 minutes, until they're cooked all the way through, and both sides are browned.
10) Remove from the grill, and serve warm or at room temperature, cut in wedges.
11) To make the caponata: In a large strainer or colander, toss the eggplant with a generous sprinkling of salt. Weigh it down, and set it in the sink to drain for 30 minutes.
12) Heat the olive oil in a large frying pan (11" to 12" pan, if possible). Add the eggplant and cook over medium heat, stirring frequently, until it's beginning to brown, about 10 minutes.
13) Stir in the onion, garlic, and tomatoes. Cover the pan, and cook gently, over medium heat, for 30 minutes, stirring occasionally.
14) Remove the cover from the pan and cook for an additional 10 to 15 minutes, or until the caponata is quite thick.
15) Add the sugar, salt to taste, and the black pepper. Remove the pan from the heat, and transfer the caponata to a bowl to cool.
16) When the caponata is lukewarm, stir in the capers, olives, and sliced plum tomatoes.
17) Serve the caponata at room temperature, with wedges of bread.
Yield: 4 flatbreads and about 4 cups caponata, about 8 servings.