Warm the oil in a large, deep skillet or Dutch oven over medium heat. Add the green curry paste and stir for about 30 seconds, until fragrant. Keep the heat moderate so the paste does not scorch.
Add the chicken and stir to coat it in the curry paste. Cook for 3 to 4 minutes, stirring occasionally, until the chicken is lightly colored on the outside.
Pour in the coconut milk and chicken broth. Scrape the bottom of the pan, then bring the mixture to a gentle simmer.
Stir in the bell pepper, zucchini, bamboo shoots, fish sauce, and brown sugar. Simmer gently, stirring once or twice, until the vegetables are tender-crisp and the chicken is cooked through, about 8 to 10 minutes.
Remove the pan from the heat. Stir in the lime juice and basil, if using. Taste carefully and add a little more curry paste, fish sauce, brown sugar, or lime juice if needed.
Serve hot with cooked jasmine rice, if desired.