Heat the oil in a 12-inch skillet over moderately high heat until hot but not smoking. Add the curry paste, lime leaves, and chile and saute, stirring occasionally, until fragrant. Add the chicken and saute, stirring occasionally, until just cooked through.
Add coconut milk, broth, brown sugar, fish sauce, and lime juice to skillet and simmer, stirring occasionally, until slightly thickened, about 5 minutes. Stir in the cilantro and scallion and simmer for about 1 minute more. Divide among 4 bowls and serve immediately with rice.