Place the chicken pieces in a medium bowl: Add the soy sauce, Shaoxing wine, cornstarch, and neutral oil. Toss until every piece is lightly coated, then let the chicken stand for 15 minutes while you prepare the vegetables and sauce. The coating should look thin and slightly glossy rather than wet or pasty.
In a small bowl, whisk together the soy sauce, rice vinegar, Shaoxing wine, chili garlic sauce, sugar, cornstarch, and chicken broth. Whisk until the cornstarch is fully dissolved. Set the sauce near the stove and stir it again immediately before using.
Cut the broccoli into small, evenly sized florets: Slice both bell peppers and the fresh chiles. Mince the garlic, grate the ginger, and separate the white and green portions of the scallions. Have everything ready before heating the wok because the stir-fry cooks quickly.
Heat a wok or large skillet over high heat until very hot. Add 1 tablespoon (15 mL) of the neutral oil and swirl to coat the surface. Add the chicken in a single layer. Cook without moving it for about 2 minutes, then stir-fry for another 3–4 minutes until browned and nearly cooked through. Transfer the chicken to a clean plate.
Return the wok to medium-high heat: Add the remaining 1 tablespoon (15 mL) oil followed by the broccoli. Stir-fry for about 2 minutes. The broccoli should turn bright green but still feel firm.
Add the red bell pepper, green bell pepper, fresh chiles, garlic, ginger, and scallion whites. Stir-fry for another 2 minutes. The peppers should soften slightly while retaining some crispness, and the garlic and ginger should smell fragrant without browning deeply.
Add the cooked chicken and any juices from the plate back to the wok. Toss with the vegetables for about 1 minute. Check a larger piece of chicken with an instant-read thermometer. It should reach 165°F (74°C) before serving.
Stir the prepared sauce again, then pour it around the edges of the hot wok. Toss continuously for 1–2 minutes. The sauce should bubble quickly, become glossy, and lightly coat the chicken and vegetables. If it becomes too thick, add 1–2 tablespoons (15–30 mL) water or chicken broth.
Add the scallion greens and toss for about 30 seconds. Remove the pan from the heat. The vegetables should remain colorful and crisp-tender, and there should be enough sauce to coat everything without pooling heavily in the bottom of the pan.
Transfer the Spicy Hunan Chicken to a serving dish. Sprinkle with the optional sesame seeds and additional scallion greens. Serve immediately while the chicken is hot and the vegetables retain their texture.