Heat olive oil in a Dutch oven over medium heat. Add onion to the hot oil and cook until soft and translucent, about 5 minutes. Add parsley and garlic; cook and stir for 2 minutes. Add shredded carrot; stir often for 1 to 2 minutes. Add cubed eggplant and saute for 2 to 3 minutes.
Pour in chickpeas and tomatoes. Add broth, oregano, thyme, and cayenne pepper. Mix and bring to a boil. Reduce heat and simmer until stew has thickened and the vegetables are soft, about 15 minutes. Add more broth if necessary. Season with salt and pepper.
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