Mix the turkey, 2 tablespoons of the soy sauce, and chili-garlic sauce in a small bowl. Marinate for 10 minutes.
In a separate medium bowl, whisk together the chicken broth, peanut butter, oyster sauce, remaining tablespoon of soy sauce, and 1 tablespoon of the lime juice; set aside.
Heat a 12-inch nonstick skillet over medium-high heat; once hot, add the oil. Add the turkey and cook, breaking up the meat with a spoon, until lightly browned and no longer pink, 5 to 7 minutes. Stir in the ginger and cook for 1 minute. Add the peanut butter mixture and simmer until thickened, about 3 minutes. Transfer the turkey mixture to a medium bowl. Stir in the coleslaw mix and remaining tablespoon of lime juice; set aside to cool for 10 minutes.
Divide the turkey mixture among the 8 lettuce leaves, and serve.