Step 1Combine the soy sauce, oyster sauce, 2 tablespoons of hoisin sauce and the cornstarch in a small bowl and set aside.
Step 2Heat the oil in a large skillet over high heat until very hot. Cook the garlic and white onion until soft, 30 seconds. Add the turkey, salt and pepper and cook the turkey until lightly browned though still slightly pink in the center, 2 to 3 minutes.
Step 3Stir in the mushrooms, bell pepper and water chestnuts. Add the sauce and continue cooking, stirring, until the sauce thickens and the turkey is cooked through, 2 to 3 minutes. Add the green onions and cook for 1 minute.
Step 4Transfer the turkey mixture to a large serving bowl. To serve, place 1/2 cup of the turkey into a lettuce leaf. Spoon some of the remaining hoisin sauce on top, then roll the top and bottom edges of the leaf toward the middle while tucking in one of the sides to make a package similar to a burrito. Repeat with the rest of the lettuce leaves and turkey mixture. Serve the lettuce cups on individual plates or seal with a toothpick and serve on a platter.