Place curry paste in a large saucepan, stir over low heat until fragrant. Add coconut milk and lime leaves; bring to the boil. Reduce heat and simmer, stirring, for five minutes.
Meanwhile, heat oil in a frying pan and cook chicken until just browned. Drain on kitchen towels.
Add chicken to the curry mixture with fish sauce, lime juice, sugar and eggplant then simmer, covered, until the eggplant is tender, about 5 minutes. Add zucchini, basil and coriander then simmer until zucchini is just tender.
Serve curry in bowls sprinkled with sliced chilli and green onions.