Remove top of eggplants and discard. Remove 3 vertical strips of skin from each eggplant, leaving it striped. Cut eggplants crosswise into ½ inch thick slices. Cover slices in cold salted water (2 tablespoons salt to 1-quart water) and soak for 30 minutes or until eggplants leaches brown juices. Rinse slices, drain and pat dry before frying.
Pour 1/2 inch of olive oil into a large cast iron skillet. Slowly heat until hot but not smoking. Fry eggplant slices in batches until golden brown on both sides. Drain on paper towels. When eggplant is cooked, remove oil and refill skillet with 1/4 of olive oil.
Add tomatoes and a pinch of sugar and fry them over medium-low heat for a few minutes, stirring often to avoid burning. Fold in the slices of eggplant with garlic and mash the mixture to puree. Continue to fry the eggplant and tomatoes in oil, stirring to avoid scorching, until all of the liquid evaporates and there is only oil and vegetables left.
Pour off the oil. Stir in paprika, cumin, hot pepper flakes and lemon juice. Let cool, cover tightly and refrigerate overnight. When serving, allow the jam to warm to room temperature.