Rinse the jasmine rice under cold water and combine with 0.5 cup water and 2 tablespoons of the coconut milk in a small pot; bring to a boil, then reduce heat to low and simmer covered for 15 minutes.
Slice the chicken breast into thin, bite-sized strips while the rice cooks.
Heat the coconut oil in a large skillet or wok over medium heat; add the green curry paste, grated ginger, and minced garlic, stirring for 1 minute until highly fragrant.
Add the chicken strips to the skillet and cook for 3-4 minutes until they are lightly browned on all sides.
Pour in the remaining coconut milk, fish sauce, and coconut sugar, stirring well to incorporate the curry paste into the liquid.
Add the sliced bell peppers and trimmed green beans to the pan, simmering for 5-7 minutes until the vegetables are tender-crisp and the chicken is fully cooked.
Remove from heat, stir in the lime juice, and serve the curry over the fluffy coconut rice, garnished with fresh cilantro.