Prep the lettuce and toppings. Separate the lettuce leaves, rinse, and pat dry.
Slice green onions and prep carrots and water chestnuts. Set aside.
Make the sauce. In a small bowl, whisk soy sauce, hoisin, rice vinegar, sesame oil, sriracha (if using), and brown sugar. Taste and adjust for sweet, salty, and heat balance.
Cook the aromatics. Heat oil in a large skillet over medium-high.
Add onion and cook 2–3 minutes until softened. Stir in garlic and ginger for 30 seconds until fragrant.
Brown the chicken. Add ground chicken and break it up with a spoon. Cook 5–7 minutes until no longer pink and lightly browned.
Season with a pinch of pepper.
Add crunch and sauce. Stir in water chestnuts and carrots. Pour in the sauce and simmer 1–2 minutes, stirring, until the mixture is glossy and slightly reduced.
Finish and garnish. Stir in most of the green onions, saving some for topping. Remove from heat.
Assemble the wraps. Spoon warm chicken mixture into lettuce leaves.
Top with remaining green onions, sesame seeds, and a squeeze of lime if you like.
Serve immediately. Arrange on a platter and let everyone build their own. Serve extra sauce or sriracha on the side.