Heat 2 tbsp oil in a heatproof casserole dish. Add eggplant and cook for 6 minutes, turning often, or until golden all over. Transfer to a bowl. Heat remaining oil and cook carrot for 5 minutes or until golden. Transfer to bowl with eggplant.
Add spices and chopped coriander to the pan. Stir for 1 minute or until fragrant. Stir in fresh and canned tomatoes and 1 cup water. Reduce heat to low, cover, and simmer gently for 5 minutes, stirring occasionally, or until flavours combine.
Return eggplant and carrot to the pan with chickpeas, stirring well. Cover and simmer for 10 minutes or until sauce thickens. Add spinach, cover, and remove from heat. Set aside for 3 minutes or until spinach wilts.
Top curry with reserved coriander leaves and serve with rice.