Wash and dry all produce. Bring 1 cup water and a pinch of salt to a boil in a small pot. Fill a medium pot with 2 inches water and bring to a simmer. Peel and mince ginger. Mince or grate garlic. Trim and thinly slice scallions, keeping whites and greens separate.
Add rice to small pot. Cover, reduce to a simmer, and cook until tender, 15-20 minutes. Keep covered.
Heat a large drizzle of oil in a large pan over medium-high heat. Pat chicken dry with paper towel and season all over with salt and pepper. Add to pan and cook until browned, 5-7 minutes. Remove from pan and set aside. Pour out and discard any fat released by chicken.
Add balsamic vinegar and soy sauce to pan, scraping up any browned bits stuck to bottom. Stir in 1 cup water, honey, sriracha, garlic, ginger, and scallion whites. Bring to a boil, then return chicken to pan and reduce to a simmer. Cover and cook until thickened, about 15 minutes. Flip chicken over halfway through.
Remove lid from pan with chicken and let sauce reduce to a sticky glaze, about 5 minutes. Meanwhile, place broccoli in medium pot of water. Increase heat to high and steam broccoli until tender, 3-5 minutes. Drain.
Toss chicken to evenly coat in glaze. Fluff rice with a fork. Divide rice and broccoli between plates, then top with chicken. Drizzle any remaining glaze from pan over plate. Garnish with sesame seeds and scallion greens.