Heat oil in a wok or large pot over medium-low heat. Add curry paste; cook and stir until fragrant, about 5 minutes. Pour in coconut milk; bring to a boil. Add chicken; cook until almost cooked through, about 5 minutes. Stir in palm sugar, fish sauce, and lime leaves. Simmer until sauce flavors combine, about 5 minutes.

Ladle chicken and sauce into a large bowl. Garnish with red chile peppers and basil.