Heat the oil in a large skillet over high heat. Add chicken and cook, stirring occasionally, until browned on all sides, about 3 minutes. Using a slotted spoon, remove the chicken and set aside in a bowl.
Reduce the heat to medium-high and add the onion, garlic, and ginger. Cook, stirring constantly, for 1 minute. Stir in the curry paste and stir for 1 minute. Stir in the coconut milk, fish sauce, and brown sugar. Bring to a simmer over medium heat. (Do not let the coconut milk come to a hard boil, or it can curdle.)
Once simmering, return the chicken to the skillet and add the green beans and bell pepper slices. Continue simmering until the chicken is cooked through and the vegetables are just tender, about 8 minutes.
Remove from the heat and stir in basil leaves and lime juice. Serve immediately over rice with chopped cilantro or more basil leaves.
