Make the dressing: In a small bowl, whisk the lime juice, lemon juice, fish sauce and honey. Set aside.
Make the larb: In a large skillet, heat the oil over medium heat. Add the onion, shallots, lemongrass, chile, and salt to taste. Cook until the vegetables begin to soften, about 5 minutes. Add the turkey and season with salt. Cook, stirring frequently, until the meat and vegetables are cooked through, about 5 minutes. Add the dressing to the pan and cook 2 minutes. Remove from the heat and stir in the mint. Season with salt and pepper.
Spoon the turkey mixture onto the lettuce leaves and arrange on a serving platter. Serve with sticky rice.