In a non-stick large skillet, cook the carrots in the water over high heat for 3-5 minutes, stirring occasionally.
Remove the carrots and reduce the heat to medium.
Cook the turkey in the same skillet until it becomes crumbly and is no longer pink.
Add the carrots, mushrooms, water chestnuts, minced garlic, minced ginger, teriyaki sauce, peanut butter, Japanese sesame oil, rice vinegar, and hoisin sauce.
With heat set to high again, cook for about 4 minutes, stirring constantly.
Add green onions, stirring constantly again for about 1 minute.
Spoon the heated mixture onto lettuce leaves and roll up.