Preheat a grill for cooking over medium heat.
Trim the stem ends off the eggplants and slice each eggplant lengthwise into 4 thick slabs, yielding a total of 8. Season both sides of each eggplant slab with salt and let sit for 30 minutes to draw out moisture.
Meanwhile, lay out a large sheet of heavy-duty foil or a double layer of regular foil. Put the string beans in the center. Drizzle with olive oil, sprinkle with salt and pepper and toss to coat. Bring the short ends of the foil together and fold twice to seal; fold in the other sides to seal, leaving room for steam. Place the foil packet on the grill, cover and cook until no longer raw but still crisp, 3 to 4 minutes. Open carefully to avoid the hot steam and let cool completely.
Whisk together the red wine vinegar, 1/4 cup olive oil, 1/2 teaspoon salt and a few grinds of pepper in a large bowl. Add the tomatoes, jalapeno, and cooled string beans. Stir gently and set aside to marinate.
Whisk together the yogurt, garlic, remaining 1 tablespoon oil, 1/2 teaspoon salt and some pepper in a medium bowl. Set aside.
Pat each eggplant slab very dry with paper towels. Brush both sides of each slab generously with olive oil. Grill the slabs, covered, flipping halfway through, until the eggplant is soft but still holds its shape, 12 to 15 minutes.
When ready to serve, add the basil, cucumbers and 1/2 teaspoon salt to the string bean-tomato salad. Drizzle some yogurt sauce over each eggplant slab and top with the salad.
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