Whisk the sauce: Combine low-sodium soy sauce, oyster sauce, chicken broth, sugar, rice vinegar, sesame oil, and 1.5 tbsp cornstarch until smooth.
Marinate the chicken: Toss sliced chicken with 1 tsp soy sauce and 1 tsp cornstarch. Set aside while the pan heats.
Heat a large skillet or wok over medium-high. Swirl in 1 tbsp oil until shimmering.
Sear the chicken in two batches until just cooked and lightly golden, 3–4 minutes total. Transfer to a plate.
Add remaining 1 tbsp oil. Stir-fry mushrooms until browned, about 3 minutes. Add zucchini; cook 2 minutes.
Stir in garlic and ginger; cook until fragrant, 30 seconds.
Return chicken to the pan. Whisk the sauce again and pour in. Cook, stirring, until thick and glossy, 1–2 minutes.
Season with black pepper. Thin with a splash of broth if needed. Garnish with green onions and serve.