Rinse the jasmine rice under cold water until the water runs clear.
In a small saucepan, combine the rice with 0.5 cup water and half of the coconut milk; bring to a boil, then reduce heat to low, cover, and simmer for 15 minutes.
While the rice cooks, slice the chicken breast into thin bite-sized strips.
Heat the avocado oil in a large skillet over medium heat and sauté the chicken until lightly browned on all sides.
Add the green curry paste to the skillet and stir for 1 minute until highly aromatic.
Pour in the remaining coconut milk, fish sauce, and coconut sugar, stirring to create a smooth sauce.
Add the snap peas and sliced red bell pepper to the skillet and simmer for 5 minutes until the vegetables are tender-crisp and the chicken is fully cooked.
Fluff the coconut rice with a fork and portion it into a bowl.
Top the rice with the green curry chicken and vegetables, then garnish with fresh cilantro and a squeeze of lime.