Place all the sauce ingredients in a small bowl and whisk well to combine. Set aside.

Add ¾ cup water to a 12-inch nonstick skillet and place it over HIGH heat. Add the broccoli and bring the water to a boil, then cover the pan, reduce the heat to MEDIUM-LOW, and steam the broccoli until crisp-tender, about 2 to 3 minutes. Drain and transfer to a bowl and cover to keep it warm. Wipe the skillet clean.

Add the oil to the dry empty skillet and place it over MEDIUM-HIGH heat. When the oil is hot, add the chicken and spread it out in an even layer. Season with salt and pepper, to taste, then cook, stirring occasionally, for about 5 to 6 minutes, or until browned and cooked through.

Reduce the heat to MEDIUM. Whisk the sauce once more and add it to the skillet. Cook, stirring occasionally, until the sauce is bubbling, then reduce the heat to LOW and continue to cook until thickened. If the sauce thickens more than desired, add a small amount of water. Remove from the heat.

Divide the rice, chicken, broccoli, and avocado between 4 to 5 bowls. Season the avocado with a pinch of salt and pepper. Drizzle the chicken with a little of the sauce from the pan and sprinkle with sesame seeds.