In a small bowl, whisk together sesame oil, soy sauce, honey, broth, cornstarch and black pepper. Set aside.
Preheat cast iron skillet or wok on medium-high heat and add 1 tablespoon of oil. Add chopped chicken breasts and cook uncovered for 7 minutes, turning once. Remove onto a plate and set aside.
Return skillet or wok to medium-high heat. Add remaining 1 tablespoon of oil, garlic and ginger. Cook for 30 seconds, stirring frequently. Add broccoli and cook for another 2 minutes.
Whisk sauce well (cornstarch settles fast at the bottom) and add to the skillet or wok. Stir and let sauce come to a boil, then cook for 2 minutes until it has thickened. Add chicken and stir to combine.