Peel alternating narrow strips from each eggplant, leaving the stems attached. Cut a deep lengthwise slit without cutting through. Sprinkle with half the salt and rest 20 minutes, then blot dry.
Heat half the oil in a wide skillet and brown the eggplants gently on several sides for 10 to 12 minutes until beginning to soften. Transfer to a roasting pan slit-side up.
Make the Filling:
Add the remaining oil and onions to the skillet. Cook over medium-low heat for 20 minutes until very soft. Add garlic for 2 minutes, then tomatoes, sugar, remaining salt, and pepper. Simmer 15 minutes until juicy but not watery.
Open each eggplant slit and fill generously with the onion-tomato mixture. Spoon any extra filling around the pan and add the water.
Bake and Rest:
Cover loosely and bake at 350 F (175 C) for 35 minutes. Uncover and bake 10 minutes more until completely tender and slightly collapsed.
Cool for at least 30 minutes. Finish with parsley and lemon juice and serve warm or at room temperature.
Notes
The eggplants should be fully tender and relaxed rather than firm enough to hold a perfect shape.
Refrigerate in the cooking juices for up to four days; bring near room temperature before serving.