This chilled cucumber-coconut soup gets its inspiration from Indian cuisine, which often marries the two components. Jalapeno and fresh cilantro add character to poached chicken.
Bring chicken stock to a boil in a small saucepan. Add chicken; return to a simmer, and reduce heat. Cook, covered, until chicken is cooked through, about 12 minutes. Transfer chicken to a plate, reserving stock, and let cool.
Heat olive oil in a medium saute pan over medium heat. Add onion; cook until soft and translucent, about 4 minutes. Add garlic, jalapeno, and cucumbers; cook 1 minute more. Add reserved chicken stock, and bring to a simmer; cook until cucumbers are tender, about 5 minutes. Remove from heat; let cool slightly.
Working in batches, transfer cucumber mixture to a blender or food processor, and puree until smooth. Add coconut milk and cilantro to last batch, pureeing until cilantro is very finely chopped. Transfer to a large bowl, and cover with plastic wrap; place in refrigerator until soup is completely cool. Stir in lime juice, and season with salt and pepper.
To serve, shred chicken into bite-size pieces. Ladle soup into bowls; top each with chicken, and garnish with cilantro.