Heat the vegetable oil in a large skillet or wok over medium-high heat. Add the sliced chicken and cook for about 5-7 minutes, or until browned and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the minced lemongrass, garlic, and ginger. Stir-fry for about 1-2 minutes until fragrant.
Add the sliced red bell pepper and snap peas to the skillet. Stir-fry for another 3-4 minutes until the vegetables are tender-crisp.
Return the cooked chicken to the skillet. In a small bowl, mix together the soy sauce, fish sauce, brown sugar, and lime juice. Pour this mixture over the chicken and vegetables, stirring to combine. Cook for an additional 2-3 minutes to heat through.
Serve the stir-fry over cooked jasmine rice and garnish with fresh cilantro.