Step 1 Cut out stem of peppers and discard. Cut each pepper in half lengthwise and remove veins and seeds. Fill large pot with enough water to cover peppers, but don't add peppers yet. Add 1 tablespoon salt and stir to dissolve. Bring water to boil, add peppers and place a plate that is slightly smaller than circumference of pot on top of peppers so they stay submerged. Parboil peppers 3 minutes. Remove and set aside.
Step 2 Slice garlic cloves lengthwise in small slivers. Hold slices together and chop crosswise so garlic is chopped into tiny pieces. Set aside. Cut onion lengthwise in half and place each half cut-side down on chopping board. Cut lengthwise in narrow strips, holding strips together, chop crosswise into small pieces. Set aside. Do tomatoes just the same, cutting in half lengthwise, then placing flat-side down, cutting into narrow slices and small pieces crosswise. Cut eggplant in the same way, though pieces don't have to be as tiny.
Step 3Remove leaves of oregano from stems and pile leaves together. Slice them lengthwise and then into tiny bits crosswise.
Step 4Coat skillet with oil and heat over medium-high heat. Add garlic and onion and cook over medium heat to soften, 1 to 2 minutes. Add tomatoes, eggplant, oregano and salt and pepper to taste. Stir to mix and blend thoroughly. Taste for salt and add a little more if it is too mild. Cover pan and cook over medium-low heat until very tender, 10 minutes, stirring once or twice. Uncover and cook another 3 minutes, stirring often.
Step 5Put pepper halves in oiled 13x9-inch baking dish. Using slotted spoon, fill halves with tomato-eggplant mixture. Bake at 350 degrees until heated through, 20 minutes. Peppers will remain crisp. Remove and serve hot or cold garnished with basil.