Fry the curry paste in the oil in a large skillet or wok over medium heat until fragrant. Stir the coconut milk into the curry paste and bring to a boil. Add the chicken; cook and stir until the chicken is nearly cooked through, 10 to 15 minutes. Stir the palm sugar, fish sauce, and lime leaves into the mixture; simmer together for 5 minutes. Taste and adjust the saltiness by adding more fish sauce if necessary. Garnish with sliced red chile peppers and Thai basil leaves to serve.