Put yogurt in cheesecloth and suspend. Let drain for at least 1 hour.
Meanwhile, wash and peel eggplant. Slice into ½-inch pieces, then spread out on a board. Sprinkle with salt; let stand 20 to 30 minutes. Wipe dry.
In a large frying pan, fry onion in oil until tender, then add tomatoes. Add jalapeño pepper, chili powder, black pepper, and salt to taste. Cook very slowly for 30 to 40 minutes.
Meanwhile, in another frying pan heat vegetable oil; fry eggplant slices until brown. Remove from pan; set aside.
In a baking dish, alternate layers of eggplant and tomato mixture. Add 2 to 3 tablespoons of water, cover, and simmer over low heat in a 300° oven for one hour and 10 minutes.
To serve: Add garlic and salt to taste to drained yogurt. Put half of yogurt mixture on the bottom of a warm serving dish, then arrange eggplant slices over yogurt. Top with remaining yogurt. Serve immediately with flatbread.