Whisk the soy sauce, 3 tablespoons water, honey, rice vinegar, sesame oil, garlic, ginger, 2 teaspoons cornstarch, and optional red pepper flakes in a small bowl.
Pat the chicken dry. Toss it with 1 tablespoon cornstarch and black pepper until evenly coated.
Heat the avocado oil in a large skillet over medium-high heat. Add the chicken in one layer and cook for 6 to 8 minutes, turning occasionally, until browned and the largest pieces reach 165°F (74°C). Transfer to a clean plate.
Add the broccoli and 1/4 cup water to the skillet. Cover and cook for 2 to 3 minutes. Remove the lid and allow remaining water to evaporate.
Whisk the sauce again, pour it into the skillet, and stir continuously for 1 to 2 minutes until bubbling, glossy, and thick enough to coat a spoon.
Return the chicken and its juices to the skillet. Toss with the broccoli and sauce for 1 minute. Garnish with sesame seeds and green onions, then serve with the optional grain.