In a small bowl or jar, whisk together the rice vinegar, soy sauce, sesame oil, brown sugar, minced ginger, and minced garlic until the sugar dissolves completely. Taste and adjust seasoning as needed. Set aside—this dressing will coat all the other ingredients and create the flavor backbone of the wraps. I like to make the dressing first so the flavors have time to meld while I prep the other components.
While the dressing rests, finely dice the celery, shred the carrots and cabbage, slice the green onions, chop the fresh cilantro, and crush or sliver the almonds. Drain the mandarin oranges thoroughly and gently separate any clumps. Organize all prepped ingredients within easy reach—this mise en place will make assembly fast and effortless.
In a large bowl, combine the shredded chicken, prepped vegetables from Step 2, water chestnuts, salt, and pepper. Pour the dressing from Step 1 over the mixture and toss gently but thoroughly to coat everything evenly. The chicken will absorb the flavors and the vegetables will stay crisp. I find that tossing the warm chicken with the dressing helps it absorb more flavor compared to mixing everything cold.
While the chicken salad sits for a moment to marinate, lightly toast the sesame seeds in a dry skillet over medium heat for about 1-2 minutes, stirring occasionally, until fragrant and lightly golden. Meanwhile, carefully separate the butter or Bibb lettuce leaves and gently rinse and dry them so they're ready to use as wraps.
Place a generous spoonful of the chicken salad mixture from Step 3 onto each prepared lettuce leaf. Sprinkle with the toasted sesame seeds from Step 4 for added crunch and nutty flavor. Fold or wrap the lettuce leaf around the filling like a taco or burrito, leaving the top slightly open so the colorful filling peeks through. Serve immediately while the vegetables are still crisp and fresh.