Rinse whole lettuce leaves and pat dry, being careful not tear them. Set aside.
Heat a large skillet over medium-high heat. Cook and stir turkey and cooking oil in the hot skillet until browned and crumbly, 5 to 7 minutes.
Drain and discard grease; place turkey in a bowl.
Cook and stir onion and carrots in the same skillet used for turkey until slightly tender, 5 to 10 minutes.
Stir hoisin sauce, garlic, soy sauce, vinegar, ginger, into onions.
Add water chestnuts, sesame oil, and cooked turkey; cook and stir until the onions just begin to wilt, about 2 minutes.
Serve with lettuce.