Place eggplant on a rimmed sheet pan and sprinkle with salt. Let rest for 30 minutes to draw out the water. Pat dry with a paper towel. Warm 2 tablespoons of olive oil in a large soup pot (Dutch oven) over medium-high heat. Add 1/2 of the eggplant and brown on several sides, cooking for about 5-7 minutes. Remove from the pot and reserve. Repeat with remaining eggplant, adding more oil if needed.
Return the pan to the stove with 1 tablespoon of olive oil and add onions. Cook until they begin to brown. Add garlic and cook 1 minute. Add tomato paste cumin and chili pepper. Cook 1 minute more. Add tomatoes along with reserved eggplant (saving 1/4 of eggplant for garnish). Cook 5 minutes to soften tomatoes.
Add stock and simmer for 10 minutes. Puré,e soup with a food mill or immersion blender and season to taste with salt, pepper, a drizzle of balsamic and garnish with reserved eggplant and fresh chives.
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