Mince the garlic cloves and ginger, then finely dice the mushrooms into roughly 1/4-inch pieces—this size mimics ground meat and ensures even cooking. Dice the water chestnuts into similar-sized pieces. In a small bowl, whisk together the hoisin sauce, soy sauce, rice vinegar, water, cornstarch, brown sugar, and sesame oil until the cornstarch fully dissolves and the sauce is smooth. This prevents lumps from forming when the sauce hits the hot pan. Separate the lettuce leaves carefully and set aside on a serving plate.
Heat 2 tablespoons of sesame oil in a large skillet or wok over medium-high heat. Once shimmering, add the diced mushrooms and salt, stirring frequently for 5-7 minutes until the mushrooms release their moisture and begin to caramelize slightly—this develops deep, savory flavor. Add the water chestnuts and cook for another 2 minutes to warm through and let them absorb some of the pan flavors. I like to keep the heat fairly high here to get a little color on the mushrooms, which adds complexity to the final dish.
Transfer the cooked mushrooms and water chestnuts to a plate. In the same skillet, add the ground turkey and 1 tablespoon of soy sauce, breaking the meat apart with a spoon as it cooks for 5-6 minutes over medium-high heat until no pink remains and the meat is lightly browned. In the final minute of cooking, add the minced garlic and ginger from Step 1, stirring constantly for about 30 seconds to bloom their flavors without burning them. This quick-cooking technique prevents the aromatics from becoming bitter.
Return the cooked mushrooms and water chestnuts to the pan with the turkey. Pour in the sauce mixture from Step 1 and stir everything together over medium-high heat for 2-3 minutes, stirring occasionally, until the sauce thickens and coats all the ingredients—you'll see it go from thin to glossy as the cornstarch activates. Remove from heat and fold in the basil, cilantro, and white pepper, being gentle so the fresh herbs stay bright. Taste and adjust seasoning if needed.
Spoon the warm turkey filling into the prepared lettuce leaves from Step 1, adding 2-3 tablespoons per wrap depending on leaf size. Top each wrap with sliced green onions for a fresh, sharp contrast to the rich, savory filling. Serve immediately while the filling is still warm and the lettuce is cool and crisp.