In a small mixing bowl, whisk together the soy sauce and cornstarch until no lumps remain. Add chicken stock, rice vinegar, brown sugar, and garlic and whisk until evenly combined.
Heat a pot or bowl of water in the microwave or on the stove until steaming. Add noodles and set let them soak until just tender. (Check the package instructions for the recommended timing, but reduce by one minute since noodles will continue to cook slightly more when added to the stir-fry.) When the noodles are tender, drain.
While noodles soak, toss chicken with some salt and pepper.
Heat a wok or non-stick pan over medium-high heat. Add 1 Tbsp oil and then chicken. Saute until chicken is cooked through, 4 to 5 minutes. Set chicken aside.
Return wok to medium-high heat. Add 1 Tbsp oil and then broccoli. Saute broccoli until it turns bright green and then continue cooking until broccoli is tender (if using frozen broccoli, just saute it until it's heated through and the excess water cooks off).
Add sauce to the pan and toss to coat broccoli. Continue cooking until sauce reduces slightly, 1 to 2 minutes.
Add chicken and noodles to the pan and toss everything to coat in sauce.
Remove from heat and add green onions and sesame seeds, if using. Serve immediately!