Heat 1 tablespoon oil in a large skilett over medium-high heat. Add cut-up chicken and season with salt and pepper. Cook for about 5 minutes until the chicken is browned on all sides and mostly cooked through. (You may need to add the additional tablespoon of olive oil if you have to brown the chicken in batches)
To make the sauce: I find it saves time to make this while your chicken is browning. In a small bowl, whisk together the soy sauce, apple cider vinegar, honey, cornstarch, garlic, and ginger until combined. Set aside.
Once the chicken is browned, it's time to add the broccoli - stir to combine. Continue cooking for an additional 3 minutes or until the broccoli is bright green and nearly tender.
Stir in the soy sauce mixture, and cook for an additional 1 minute until the sauce has thickened and broccoli is tender.
Remove from heat and garnish with green onions and toasted sesame seeds. Serve immediately.