Heat the vegetable oil in a large pot or skillet over medium-high heat.
Add the sliced chicken and cook until browned on all sides. Remove and set aside.
In the same pot, add the sliced onion and minced garlic. Sauté until the onion is soft and translucent, about 5 minutes.
Add the red curry paste to the pot and stir-fry for 1-2 minutes until fragrant.
Pour in the coconut milk and chicken broth, stirring to combine.
Bring the mixture to a simmer.
Add the sliced bell peppers, zucchini, and bamboo shoots to the pot.
Return the browned chicken to the pot as well.
Stir in the fish sauce and brown sugar.
Simmer the curry for about 15-20 minutes, or until the chicken is cooked through and the vegetables are tender.
Stir in the Thai basil leaves and let the curry simmer for another 2-3 minutes.
Enjoy your meal!