Wash and dry the eggplants. Peel them in alternating lengthwise strips, leaving some skin attached. Make one shallow lengthwise slit in each eggplant without cutting all the way through.
Place the eggplants in salted water for 10 to 20 minutes. Rinse and pat completely dry.
Heat 4 tablespoons olive oil in a wide skillet over medium-low heat. Add the sliced onions and a pinch of salt. Cook for 8 to 10 minutes, stirring regularly, until soft and lightly golden.
Add the garlic and green peppers and cook for 2 minutes. Stir in the diced tomatoes, optional tomato paste, black pepper, optional paprika and optional sugar. Cook for about 10 minutes, until juicy but not watery. Remove from the heat.
For the traditional method, gently fry the dried eggplants in olive oil until softened and lightly coloured on all sides, then drain on paper towels. For the roasted method, brush the eggplants with olive oil and roast at 400°F / 200°C for 25 to 30 minutes, until tender.
Arrange the softened eggplants in a wide pot or baking dish with the slit sides facing upward. Carefully open their centres with a spoon.
Divide the onion and tomato filling evenly between the eggplants. Top each one with a slice of tomato and a piece of green pepper.
Pour 120 ml hot water around the eggplants. Cover and cook over very low heat for about 20 minutes, or bake at 350°F / 180°C for 25 to 30 minutes, until very tender.
Rest for at least 20 minutes before serving. Scatter with chopped parsley and add a few drops of lemon juice if desired.
Serve warm, at room temperature or cool with rice pilaf, yogurt, cacık or fresh bread.