Heat the vegetable oil in a large pot or skillet over medium heat.
Add the green curry paste and stir-fry for 1-2 minutes, until fragrant.
Add the thinly sliced chicken breast to the pot and cook for 2-3 minutes until the chicken is no longer pink.
Pour in the coconut milk and chicken broth, stirring to combine.
Bring the mixture to a simmer.
Add the eggplant, bamboo shoots, red bell pepper, and snap peas to the pot.
Simmer for 10-15 minutes, or until the vegetables are tender.
Add the fish sauce and brown sugar, stirring to combine.
Adjust the seasoning to taste, adding more fish sauce or sugar if needed.
Stir in the Thai basil leaves and let the curry simmer for another 2-3 minutes.
Enjoy your meal!