Steam the Broccoli:
Fill a large pot with 2-3 inches of water and add the broccoli along with 1 teaspoon of salt. Use a steamer basket if you have one, but it’s not necessary. Cover and bring to a simmer. Reduce the heat to low and cook for 6 minutes, or until the broccoli is tender-crisp. Drain the broccoli in a colander and rinse with cold water to stop the cooking process.
Prepare the Sauce:
In a small bowl, stir together the soy sauce, water, sesame oil, honey, rice vinegar, grated ginger, minced garlic, cornstarch, and sesame seeds. Set the sauce aside.
Prepare the Chicken:
Place the chicken pieces in a large plastic zipper bag. In a small bowl, whisk together the cornstarch, flour, ½ teaspoon salt, and ¼ teaspoon pepper. Sprinkle the cornstarch mixture onto the chicken. Seal the bag and toss to evenly coat all of the chicken pieces.
Cook the Chicken:
Heat the olive oil in a large skillet over medium heat. Once the oil is hot (it should shimmer but not smoke), place the chicken pieces in a single layer in the skillet. Cook without touching them for 2-3 minutes, or until golden brown. Flip the chicken to cook all remaining sides until golden and fully cooked through (no pink in the center).
Cook the Sauce:
Once the chicken is cooked through and golden brown on all sides, pour the prepared sauce over the chicken in the skillet, keeping the heat on medium-low. Toss the chicken to coat it in the sauce and bring the sauce to a simmer. Stir the chicken in the sauce until it thickens, then remove the skillet from the heat.
Combine Chicken and Broccoli:
If there is room in your skillet, add the steamed broccoli to the chicken mixture and gently fold to evenly incorporate. If not, mix the chicken and broccoli together in a separate bowl. Sprinkle with diced green onions and serve warm.