Whisk together the tamari, lime juice, honey, cornstarch, white pepper, and water. Set aside the sesame oil separately. Heat the skillet over medium-high for 2 minutes.
Add 1 tablespoon oil and cook the chicken in two batches, pressing into a thin layer. Leave undisturbed for the first 2 minutes, then break into bite-size clusters. Season with ½ teaspoon salt total across both batches. Cook to 165°F (74°C) and transfer to a plate.
Add the mushrooms in a broad layer and cook 6–8 minutes, stirring occasionally, until their liquid evaporates and edges brown. Add the onion and scallion whites; cook 3–4 minutes. Add the ginger and garlic; stir 30 seconds until fragrant.
Add the carrots and cook 2 minutes. Add half the cabbage, toss until it loses some volume, about 2 minutes. Add the remaining cabbage and the last ½ teaspoon salt. Cook 4–6 minutes over medium-high, tossing from the bottom, until thick ribbons bend but still resist slightly when bitten.
Return the chicken and plate juices. Add the water chestnuts. Re-whisk the sauce, pour it around the hot edge of the skillet, and toss 2–3 minutes until the sauce is glossy and no puddle remains when food is pushed aside.
Turn off the heat. Drizzle in the sesame oil and fold in half the scallion greens and half the sesame seeds. Rest 3 minutes, then taste and adjust with a small squeeze of lime or a teaspoon of tamari if needed. Top with the remaining scallions and sesame seeds.