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Bring the coconut milk to the boil in a pan just large enough to fit the chicken in a single layer snugly.
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Add chicken and turn to coat in the coconut milk. Then place the beans on top. Simmer for 5 minutes.
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Cover the pot and remove from the heat. Stand for 20 minutes.
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When the 20 minutes are up cut into the chicken to make sure it is cooked. If there are still some pink bits, bring back to a simmer for another few minutes.
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Combine lime juice and fish sauce with 6 tablespoons of the coconut milk poaching liquid, discarding the remaining poaching liquid. Taste and season with more fish sauce, lime or coconut milk as needed.
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Toss beans in the dressing. Slice chicken finely and toss in with the beans along with the basil leaves before serving.