Bring the water to a boil in a medium saucepan. Remove from heat, and add mushrooms; let sit until softened, at least 30 minutes. Strain mushrooms through a cheesecloth-lined sieve to remove any sandy grit; discard mushrooms.
Return mushroom broth to a clean medium saucepan, and add chicken, carrot, jalapeno, ginger, garlic, and peppercorns. Bring just to a boil; reduce heat, and simmer over medium heat 20 minutes, or until broth is reduced to about 4 cups. Strain through a fine sieve back into saucepan, reserving broth and discarding everything but the chicken.
Remove chicken meat from the bones, and shred the meat. Return chicken meat to mushroom broth; add sugar, tofu, lime juice, herbs, and soy sauce, and stir to combine. Transfer soup to a covered container, and refrigerate until well chilled, at least 2 hours or overnight. Serve.
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