Place tomatoes in a fine-mesh strainer set over a bowl and toss with a generous pinch of salt. Let stand 30 minutes.
Meanwhile, heat 1/4 inch oil in a large skillet over medium-high heat until shimmering. Working in batches if necessary, fry eggplant slices, rotating for even browning and turning once halfway through, until golden on both sides and tender throughout, about 5 minutes; lower heat if oil begins to smoke. Transfer eggplant to a paper towel–lined baking sheet and sprinkle with salt.
Transfer tomatoes to a medium bowl and discard drained liquid. Add cucumber, lemon juice, and parsley to tomatoes. Season Israeli salad with salt to taste and mix well.
In a medium bowl, toss cabbage with vinegar and season with salt.
In each pita pocket, smear 3 tablespoons (45ml) of hummus in an even layer. Layer 2 to 3 slices fried eggplant on top of each. Drizzle each with 1 tablespoon (15ml) tahini sauce, then top with sliced eggs, Israeli pickles, and remaining tahini sauce. Spoon some of the Israeli salad into each pita, drizzle with amba, and serve.