In a small saucepan, bring two cups of salted water to a boil. Add the broccoli and cook for 3 minutes. Drain and set aside.
Season the chicken pieces with salt and pepper. Dredge each in the flour, shake off the excess, and transfer it to a colander. In a large skillet over medium-high heat, warm 2 tablespoons oil. Add the chicken and brown until golden and cooked through, about 4 minutes per side. Transfer it to a plate.
In the same skillet (don't wipe it out), warm the remaining oil. Add the onion, garlic, and ginger, and cook until soft, about 3 minutes. Add the broccoli and cook until heated through, about 3 minutes. Stir in the broth, juice, and soy sauce, scraping up the browned bits at the bottom of the pan. Bring the mixture to a simmer, and let it cook and reduce for 2 minutes.
Add the chicken and cook until heated through, about 3 minutes. Transfer to a serving platter and sprinkle with the cilantro. Serve with rice.