Heat the vegetable oil in a large skillet over medium heat.
Add minced garlic and ginger; sauté for about 30 seconds until fragrant.
Add the ground chicken to the skillet, breaking it up with a spoon. Cook until browned and no longer pink, around 5-7 minutes.
Stir in the diced red bell pepper, shredded carrot, chopped green onions, and chopped water chestnuts. Cook for an additional 2-3 minutes.
In a small bowl, whisk together soy sauce, fish sauce, hoisin sauce, and lime juice. Pour this mixture into the skillet with the chicken and vegetables, stirring to combine. Let it cook for another minute.
Remove the skillet from heat and allow the mixture to cool slightly.
Lay out the lettuce leaves on a serving platter. Spoon the chicken mixture into each leaf, making sure to fill them generously.
Garnish with fresh cilantro and sliced jalapeños, if desired.