Heat one-third of the coconut milk in a large pan over medium heat until thick and oil begins to separate.
Stir in red curry paste and cook until fragrant, about 20-30 seconds.
Add remaining coconut milk, fish sauce, and palm sugar; stir to combine.
Add chicken pieces and simmer until cooked through, approximately 8 minutes.
Add bamboo shoots and kaffir lime leaves; continue simmering for 5-8 minutes.
Remove from heat and stir in Thai basil leaves and sliced red peppers.
Let curry rest for 5 minutes before serving to allow flavors to meld.