Heat the vegetable oil in a large skillet or wok over medium-high heat until shimmering. Add the green curry paste and stir constantly until highly fragrant, about 1 to 2 minutes.

Add the sliced chicken breast to the skillet and toss to coat it evenly in the curry paste. Cook for 3 to 4 minutes until the chicken is lightly browned on the outside but not fully cooked through.

Pour in the coconut milk and chicken broth, stirring well to dissolve any clumps of curry paste into the liquid. Bring the mixture to a gentle simmer.

Stir in the bamboo shoots, zucchini, and red bell pepper. Reduce the heat to medium-low and let the curry simmer uncovered for 8 to 10 minutes, or until the vegetables are tender-crisp and the chicken is fully cooked through.

Stir in the fish sauce and brown sugar, tasting and adjusting the seasoning if necessary to balance the salty and sweet flavors. Remove the skillet from the heat.

Gently fold in the fresh basil leaves until they are just wilted, which should take about 30 seconds. Serve the hot curry immediately over warm steamed jasmine rice with a wedge of fresh lime on the side.