Make the sauce: In a small bowl, combine hoisin sauce, soy sauce, water, rice wine vinegar, toasted sesame oil, and sriracha or gochujang if using. Set aside.
Cook the meat: Heat olive oil in a large cast-iron skillet or heavy skillet over medium-high heat. Add the ground chicken or turkey and cook for 3 to 4 minutes, breaking it up as it cooks, until just cooked through. Add the diced onion and cook for 3 to 5 minutes, until the onions are softened and the chicken is browned.
Add aromatics: Stir in the garlic and freshly grated ginger and cook for about 30 seconds, just until fragrant.
Add the sauce: Pour the prepared sauce into the skillet and sauté for 1 minute, stirring to coat the meat evenly.
Finish the filling: Stir in the diced water chestnuts and green onions. Cook for 1 more minute. Season with kosher salt and fresh black pepper to taste.
Serve: Spoon the warm chicken mixture into lettuce cups and garnish with the remaining green onions.
