Prep the chicken. Pat the chicken dry and slice it thinly across the grain.
Toss with 1 tablespoon cornstarch and a pinch of salt and pepper. This helps the chicken sear quickly and stay tender.
Make the sauce. In a bowl, whisk soy sauce, honey, rice vinegar, toasted sesame oil, broth, and chili-garlic sauce if using. In a small cup, mix the remaining 1 tablespoon cornstarch with 2 tablespoons water to make a slurry.
Set both aside.
Heat the pan. Place a large skillet or wok over medium-high heat. Add 1 tablespoon neutral oil and let it get hot until shimmering.
Sear the chicken. Add chicken in a single layer. Cook without moving for 2 minutes, then stir-fry until just cooked through, 3–4 minutes more.
Transfer to a plate.
Stir-fry the veggies. Add another teaspoon of oil if the pan looks dry. Toss in broccoli, bell pepper, carrot, and snap peas. Stir-fry 3–4 minutes until crisp-tender.
Add garlic, ginger, and the white parts of the green onions; cook 30 seconds until fragrant.
Build the sauce. Return chicken to the pan. Pour in the sauce and bring to a simmer. Stir the cornstarch slurry and drizzle it in while tossing.
Cook 1–2 minutes until the sauce turns glossy and clings to the chicken and vegetables.
Finish and season. Stir in the green parts of the green onions and 1 tablespoon sesame seeds. Taste and adjust with a splash more soy (salt), a squeeze of honey (sweetness), or a dash of vinegar (brightness). Add a few drops of sesame oil if you want extra nuttiness.
Serve. Spoon over steamed rice or toss with noodles.
Sprinkle with more sesame seeds and serve with lime wedges if you like a citrus pop.