Prep: Slice chicken thinly, grate ginger, mince garlic, julienne carrot, slice bell pepper, and separate scallion whites/greens. Mix sauce: whisk soy sauce, mirin, brown sugar, honey (if using), sesame oil, rice vinegar, fish sauce (if using), garlic, and ginger in a bowl. Set aside.
Cook noodles: Boil a large pot of salted water. Cook noodles 1 minute shy of package directions for al dente. Reserve ¾ cup of noodle water, drain, and toss noodles with a drizzle of oil to prevent sticking. Set aside.
Dry & season chicken: Pat chicken dry with paper towels. Season lightly with salt and pepper.
Sear chicken: Heat a large wok or sauté pan over high heat until hot. Add 1–2 tbsp neutral oil. Add chicken in a single layer (work in batches if necessary). Sear without moving 1–2 minutes to brown, then toss and cook until just done (thighs ~4–6 min, breasts ~3–5 min). Remove chicken to a plate.
Sauté vegetables: Reduce heat to medium-high. Add a small splash of oil if needed. Sauté scallion whites, carrot, and bell pepper 2–3 minutes until bright and lightly tender. Add greens (bok choy/spinach) and cook until wilted.
Add chicken & sauce: Return chicken to the pan. Pour in the prepared teriyaki sauce and bring to a simmer. Stir the cornstarch slurry, then whisk into the simmering sauce; it will thicken into a glossy glaze. If too thick, loosen with reserved noodle water, 1–2 tbsp at a time.
Toss noodles: Add drained noodles and toss thoroughly to coat. Cook 1–2 minutes so flavors marry and noodles heat through.
Finish: Turn off heat. Drizzle a little toasted sesame oil, scatter green scallion slices and toasted sesame seeds. Add a squeeze of lime if desired.
Serve: Plate immediately in shallow bowls, garnish with extra scallions, sesame seeds, and chili oil or lime wedges at the table.