Step 1Grill the eggplants over a flame or under the broiler, turning them with tongs until charred on all sides, about 15 minutes. While still hot, plunge them into a plastic bag, close tightly, and let them steam for about 15 minutes to loosen the skin.
Step 2Scrape the skin from the flesh and shake the pulp in a colander to remove excess liquid.
Step 3Puree the garlic with the eggplant pulp in a food processor. Add 3 cups of vegetable broth, process and then strain the contents into a saucepan. Reheat the soup gently over medium-low heat. If you want a thinner soup, adjust the consistency by adding more broth if necessary, and season with salt and pepper to taste. Add the vinegar and sugar.
Step 4Spoon a ladleful of hot soup into the bottom of serving bowls, cover with dollops of yogurt, and continue to layer in this manner.