In a small saucepan, combine broth, soy sauce, lime juice, sesame oil, garlic, ginger, red pepper flakes, and pepper to make the sauce. Bring the mixture to a gentle simmer over medium-low heat, stirring often to prevent boiling. Simmer for about 20 minutes until the sauce thickens, reduces by 2/3, and coats the back of a spoon. Remove from heat and let it cool for a minute.

While the sauce simmers, season sliced chicken with salt and pepper. Heat a large sauté pan over medium heat, add oil, and let it become hot. Cook chicken slices for 30-60 seconds per side, ensuring not to overcrowd the pan. Repeat in batches if necessary. Retain the chicken juices in the pan, add broccoli, and cook for 3-5 minutes until tender. If needed, add a splash of broth to aid in steaming. Reintroduce the chicken to the pan, avoiding the chicken juices, and mix to combine.

Pour in the prepared sauce, ensuring everything is well-coated. Mix thoroughly. Top with sesame seeds and green onions, if desired. Serve the chicken and broccoli stir-fry with your preferred sides. Enjoy!