Preheat oven to 400 degrees F. Wash eggplants; trim and cut into 1/2- to 1-inch pieces. Divide eggplant between 2 15x10x1-inch baking pans or shallow roasting pans. Use a sharp knife to cut off the top 1/2 inch from garlic bulbs to expose individual cloves. Leaving garlic bulbs whole, remove any loose, papery outer layers. Place garlic bulbs on a 12-inch square of heavy foil; drizzle with 1 tablespoon of the olive oil. Wrap foil up around the garlic bulbs to completely enclose.
Drizzle 1/4 cup of the remaining olive oil over eggplant in each pan and sprinkle with salt; toss to coat. Place eggplant and garlic packet in oven, placing pans on 2 separate oven racks. Roast, uncovered, 20 minutes, stirring once. Add half of the sweet pepper to each pan with eggplant; stir to combine. Roast, uncovered, 20 minutes more or until vegetables are tender, stirring once. Remove pans and garlic from oven and cool.
Transfer eggplant and peppers to a very large glass or nonreactive bowl. Squeeze garlic pulp from individual cloves into a small bowl; use the back of a spoon to mash the pulp. Add remaining oil and lemon juice to garlic; whisk to combine. Add oil mixture, parsley, and snipped oregano to eggplant mixture; toss to combine.
Serve immediately, or let mixture stand, covered, up to 1 hour before serving. Or cover and chill mixture up to 3 days; let stand at room temperature 30 minutes before serving. (Or, transfer mixture to a freezer container. Cover and freeze up to 3 months. Thaw in refrigerator before serving, then let stand at room temperature for 30 minutes.) Garnish with oregano leaves. Serve with flat breads pieces. Makes about 40 (1/4-cup) servings.