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Seed the tomatoes. Brush them and the pepper with oil.
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Light a medium-sized fire in the barbecue. When the coals are red and dusted with ash, spread them in a tight, single layer; they need to emit concentrated heat.
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Place a handful of grape vine cuttings or wood chips over the coals and let them burn for a minute or two, then set the grill over the coals and put the barbecue cover in place, vents open. When the grill is hot, place the vegetables directly over the coals, cut-side down. Grill until the tomatoes and the peppers are marked with grill marks, 2 to 3 minutes, then turn them, sprinkle with salt, cover the grill, and cook until the tomatoes are tender, about 4 minutes. Transfer the tomatoes from the grill to a shallow soup bowl or platter that will catch the juices. Cover the grill and continue grilling the peppers until their skin is blackened and the peppers are tender, an additional 5 minutes. Transfer the peppers to a paper bag, to sweat.
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Remove the skin from the tomatoes and the peppers, and discard. Save all the juices from both the tomatoes and the pepper.
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Place the tomatoes with their juices, all but one quarter of the pepper with any juices from the pepper, the onion, the cucumber and the garlic in the work bowl of a food processor fit with a steel blade, and process until you have a thick purée. Add the lemon juice and the piment d’Espelette, pulse, and adjust the seasoning. Place the gazpacho in the refrigerator, covered, to chill. This will take at least 2 hours.
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About 10 minutes before serving the soup, remove it from the refrigerator. Dice the remaining quarter of grilled pepper. Adjust the seasoning of the soup and divide it among 6 or 8 bowls. Drizzle each bowl with a bit of the remaining olive oil, garnish with some of the diced pepper and a basil leaf or two.