Warm the olive oil in a large skillet over medium-high heat. Add the ground turkey along with the minced garlic and ground ginger, cooking and breaking up the meat until browned and cooked through, about 6 minutes.
Transfer the cooked turkey to a mixing bowl, then stir in the sliced green onions and chopped water chestnuts thoroughly.
In a separate small bowl, whisk together the hoisin sauce, soy sauce, rice vinegar, and roasted red chili paste. Pour this sauce over the turkey mixture and toss everything to coat well.
Spoon about a quarter cup of the turkey filling into each lettuce leaf. Serve immediately and enjoy your flavorful wraps!