In a large saucepan over medium heat, heat the olive oil. When the oil is hot, add the onions and saute until soft, about 5 minutes. Add the garlic and bay leaves and cook for 5 more minutes. Add the tomatoes, 1 tablespoon salt, and thyme and cook for 15 minutes or until the tomatoes soften. Add the water and stock and bring to a boil. Reduce heat to a simmer and continue to cook until the tomatoes are very tender, about 10 minutes. Remove the bay leaves. Using a handheld blender, puree the soup until smooth. Chill the soup. Add 1 tablespoon of the lemon juice and hot sauce to taste, as desired.
While the soup is chilling, in a mixing bowl, combine the lobster tails, mango, papaya, pineapple, lime juice, remaining 1 tablespoon of lemon juice, extra-virgin olive oil, oregano, parsley, scallions, chives, remaining 1/4 teaspoon salt and the pepper and toss gently to thoroughly combine.
When ready to serve, ladle the soup into individual bowls. Float a lobster claw in the center of the soup and divide the lobster garnish among the bowls, about 3 tablespoons per bowl. Garnish each bowl with some of the creme fraiche and a basil sprig.
Cook's Note: This soup can be served either chilled or warm.
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