Bring ½ cup coconut milk to a boil in a heavy saucepan.
Whisk in the curry paste and cook until fragrant, about 5 minutes.
Add the chicken and cook until browned, about 5 minutes.
Stir in the 2 cups coconut milk, fish sauce, sugar, vegetables, bamboo shoots, and basil; simmer until the chicken is cooked through, about 15 minutes.
Pour the lime juice over the dish and remove from heat.
Serve warm.
