Cut chicken into 1-inch cubes. Chop broccoli into small florets. If using frozen broccoli, thaw and drain well.
Heat 1–2 tablespoons olive oil in a large skillet over medium-high heat. Season chicken with salt and pepper and cook until browned and cooked through, about 5–6 minutes.
In a small bowl, combine 3 tbsp reduced-sodium soy sauce, 2 tbsp apple cider vinegar, 2 tbsp honey, 1 tbsp cornstarch, 1 minced garlic clove, and ½ tsp ground ginger. Set aside.
Stir broccoli into the skillet with chicken. Cook 2–3 minutes until just tender and bright green.
Pour sauce over chicken and broccoli. Stir and cook for 1–2 minutes until sauce thickens. Sprinkle with 1 tsp toasted sesame seeds before serving.