Cook the jasmine rice according to the package directions and keep it warm.
Heat the oil in a 12-inch / 30 cm skillet or wok over medium heat.
Add the green curry paste and fry for 1 minute, stirring, until fragrant and slightly darker.
Pour in half the coconut milk and stir to combine. Simmer for 2 minutes until it thickens slightly and the oil starts to separate at the edges, this is normal.
Add the remaining coconut milk, vegetable broth, brown sugar, and soy sauce. Bring to a gentle simmer.
Add the broccoli and simmer for 3 minutes.
Add the bell pepper, zucchini, and bamboo shoots. Simmer for 4 minutes until the vegetables are crisp-tender.
Add the snap peas and simmer 2 minutes more.
Remove from the heat and stir in the lime juice and Thai basil leaves until the basil wilts.
Taste and adjust with more soy sauce or sugar if needed. Serve over the rice, topped with sliced chili and extra basil.