Preheat the oven at 200°C (390°F).
Roast the aubergine. Wash and pierce the aubergine skin all around with a fork and roast for about 3-4 minutes per side on a very hot pan. Next transfer them into a tin and bake for about 30 to 40 minutes, turning them once. After this step, place them into a plastic bag while they're still hot (mind your fingers!) and allow to chill at room temperature (ideally) overnight or if you're in a rush for 30 minutes in the freezer.
Wash the peppers, then cut them in half and remove their seeds. Place on a tray, coat with some olive oil and a pinch of salt, and bake for 20-30 minutes on grill mode until they are slightly brown on both sides, turning them half way.
In the meantime, chop the onion and the garlic for the tomato sauce, and allow them to sweat in a wok with some olive oil and a splash of water for about 5 minutes.
Add chopped tomatoes, tomato paste, chickpeas, sumac and season with pepper and salt. Add some extra water (up to 1 cup) and cook on medium heat for about 15 minutes.
In the meantime peel off the skin of the aubergines, shred the pulp into fillets and set aside.
As soon as the green peppers are ready, add them into the wok, stir well and cook altogether for another 15 to 20 minutes.
Combine the aubergines fillets during the last 5 mins of cooking.
Serve the Lebanese Moussaka at room temperature or cold with some fresh parsley or mint leaves finely chopped on top.