Prep the lettuce and aromatics. Gently separate the lettuce leaves, rinse, and pat dry. Set aside in the fridge to keep them crisp.
Dice the onion and bell pepper, mince the garlic, and grate the ginger.
Make the sauce. In a small bowl, whisk soy sauce, rice vinegar, hoisin (or honey+miso), and sriracha if using. Stir in the cornstarch slurry until smooth. This will help the filling cling together.
Brown the turkey. Heat a large skillet over medium-high heat.
Add sesame oil, then the ground turkey. Season lightly with salt and pepper. Cook, breaking it up with a spatula, until no longer pink and lightly browned, 5–7 minutes.
Add aromatics and veggies. Stir in onion, bell pepper, mushrooms, garlic, and ginger.
Cook 3–4 minutes, until the onion softens and the veggies release their moisture. If using carrots or water chestnuts, add them now.
Simmer with sauce. Pour the sauce over the turkey mixture. Stir and cook 2–3 minutes, until the liquid thickens and coats the filling.
Adjust seasoning with extra soy sauce or a pinch of salt if needed.
Finish and garnish. Remove from heat and stir in most of the green onions, reserving a few for topping. If you like a brighter finish, squeeze in a little lime juice.
Assemble the wraps. Spoon the warm turkey filling into chilled lettuce cups. Top with remaining green onions, sesame seeds, cilantro, or nuts.
Serve with lime wedges.