Cut the zucchini in half lengthwise and then crosswise into 1-inch chunks; set aside. In a medium saucepan or heavy skillet, bring 3/4 cup of the coconut milk to a gentle boil over medium-high heat. Cook for 2 to 3 minutes, until it begins to thicken and become fragrant Add the curry past and cook 2 to 3 minutes, pressing and stirring to desolve it into the coconut milk. Add the chicken and cook 2 minutes more, tossing to coat it with the sauce.

Add the remaining 3/4 cup coconut milk, the chicken broth, zucchini, fish sauce, and palm sugar and bring to a gentle boil. Reduce heat to maintain a lively simmer and cook, stirring now and then, until the chicken is cooked and the zucchini is tender but still firm, 8 to 10 minutes. Remove from the heat and serve hot or warm.