Make the Gazpacho: In small batches, mix all ingredients in a blender until very smooth. Pay attention to the consistency. You may have to add more water, as the water content in the ingredients may vary. Strain and chill.
Make the Lobster and Dressing: Fill a large pot with water and add plenty of salt. Bring to a boil and add lobsters. Cook for 1 minute. Remove from water, drain and chill.
Once the lobsters are cool, take off the head and remove the coral and liquid. Pass the coral through a chinois or fine-mesh sieve. Set aside the resulting liquid for use in the dressing. Peel the lobster tails and cut each tail into 6 medallions. Carefully crack the claws and remove the meat. The idea here is to keep the claw meat whole. Split the claw meat in half lengthwise. Refrigerate the lobster meat until needed.
In a bowl, whisk together the reserved coral, vinegar, and oil until smooth and blended. Season with salt, to taste. Set aside.
Make the Garnish: Cut the ends off the tomatoes. Cut the tomatoes in half lengthwise. Peel back the skin and flesh to expose the seeds. Remove the seeds, taking care to keep the mass of seeds whole. The point here is to remove the tomato seeds and their surrounding gel intact. Set aside. (Reserve the tomato flesh for another use.)
Cut the cucumber in half lengthwise and remove seeds. Sprinkle the cucumber generously with salt and let sit for an hour, in a colander in the sink. (The salt will cause the cucumber to release water.) Rinse the cucumber and pat dry with paper towels. Cut the cucumber into a tiny dice.
In a bowl, combine the cucumber, peppers, and shallot. Set aside.
Pour the oil for frying into a large saucepan to a depth of 2 inches. Heat over medium heat until a deep-fry thermometer inserted in the oil reads 375 degrees F. Fry the bread cubes until golden brown. Using a slotted spoon, transfer the croutons to a paper towel-lined plate to drain.
To Serve: Pour the chilled gazpacho into a pitcher.
In the center of each bowl, place 2 medallions and 1 claw half of lobster, and 1 tomato seed "filet." Arrange the some of cucumber mixture around the edge of the bowl, sprinkle with chives, and top with 4 croutons. Drizzle the dressing around the lobster, and drizzle the lobster with some of the extra-virgin olive oil. Finally, sprinkle everything lightly with the fleur de sel. At the table, set the bowls in front of your guests and pour some of the gazpacho into each.