To cook the tapioca pearls, pour the water into a saucepan and bring to a boil over high heat. Add the tapioca pearls, reduce the heat to a very gentle simmer, and cook for 45 minutes to 1 hour, or until all the pearls are translucent. Don't worry if some break into pieces. Drain in a colander, rinse under cold water, and transfer to a bowl . You should have 2 cups.
To make the soup, in a saucepan, heat the oil over low heat. Add the onion, tomato, carrot, leek, lemongrass, ginger, kaffir lime leaves and garlic and cook, stirring occasionally, for 5 to 6 minutes, or until the onions are soft and translucent. Add the curry powder, mix well, and continue to cook for about 30 seconds. Add the chicken broth and simmer over low heat for 8 to 10 minutes, or until the vegetables are soft. Add the coconut milk, vinegar, and sugar, raise the heat to medium-high, and bring to a boil. Remove from the heat and carefully place some of the vegetables and liquid in a blender. Process very slowly to a smooth, silky puree. Pour through a fine-mesh sieve placed over a saucepan, forcing the liquid through the sieve with a rubber spatula. Repeat in small batches with the remaining vegetables and liquid. Season with salt and pepper and reheat over low heat to serving temperature.
To prepare the garnishes, put the shrimp in a microwave-safe bowl, season with salt and pepper, and add the butter. Cover with plastic wrap and microwave for about 2 minutes on full power (the timing will depend on the microwave), or until the shrimp are pink and opaque throughout. Keep warm. To serve, have ready sixteen 1/2-cup demitasse or sake cups. Thread 1 shrimp, 1 mango slice and 1 cilantro sprig onto each of 16 decorative picks. Put 1/4 cup of the hot soup into each cup, and then add 2 tablespoons tapioca pearls. Place a skewer on the rim of each cup and serve immediately.
Photos from the Great Gatherings: Star Chefs Entertain at Home Cookbook