Prepare the eggplant. You may do this on the grill (recommended), stovetop, or oven.
Preheat the grill to high heat. Place whole eggplants directly on coals or (if using gas) on the grates.
Carefully cover your stovetop with foil--this is gonna get messy. Place whole eggplants directly over a high flame.
Preheat broiler. Place eggplants on a baking sheet on the top rack of oven.
ALL THREE METHODS:
Watch eggplants carefully as they cook, turning occasionally, until they are completely charred on all sides and collapsing. Remove from heat and place in bowl. Cover and let cool.
When eggplant has cooled, carefully scrape the flesh from the skin, discarding all charred bits. Place the flesh in a blender and add chicken stock, yogurt, and salt. Blend until smooth. Taste and add salt if necessary. Chill soup.
Spoon into individual bowls before adding drizzle of olive oil, a pinch of red chile flakes, and a generous helping of za'atar.