In a small bowl, whisk together soy sauce, oyster sauce, honey, cornstarch, chicken broth, and sesame oil until smooth, then set aside.
Heat 2 tablespoons of vegetable oil in a large skillet or wok over high heat until very hot.
Add the chicken pieces in a single layer and cook for 4-5 minutes without stirring, then flip and cook another 3-4 minutes until golden brown and cooked through. Remove chicken to a plate.
Add the remaining 1 tablespoon oil to the same skillet, add broccoli florets and stir-fry for 3-4 minutes until bright green and crisp-tender.
Add minced garlic and cook for 30 seconds until fragrant, then return the chicken to the skillet.
Pour the sauce over everything, toss well, and cook for 2-3 minutes until the sauce thickens and coats the chicken and broccoli. Serve immediately over rice.