Drain chickpeas and set aside. Heat oil in cooker over medium-high heat. Add onion and cook 2 minutes, stirring frequently. Add water, reserved chickpeas, bulgur, caraway seeds, and red pepper flakes. Set eggplant on top.
Lock lid in place. Over high heat bring to high pressure. Lower heat just enough to maintain high pressure and cook for 16 minutes. While stew is cooking, blend tomato paste and tahini into hot water. Set aside.
Make the Tomato-Cucumber Salad: In a large bowl, mix tomatoes, cucumbers, lemon juice, cilantro, and salt. Set aside.
After 16 minutes, quick-release pressure. Remove lid, tilting it away from you to allow any excess steam to escape. Set cooker over medium heat. Stir in tomato paste mixture, garlic, 2 teaspoons coriander and salt to taste.
Stir well. If necessary, mash eggplant pieces against the sides of cooker and blend them in to create a thick, creamy sauce. Add more red pepper flakes and additional coriander, if needed. Cook over medium heat until garlic loses its raw edge, about 3 minutes. Ladle stew into large, shallow bowls and sprinkle on a liberal portion of the Tomato-Cucumber Salad. Pass the remaining salad at the table.