Place the sliced flank steak in the bottom of the slow cooker. In a mixing bowl, whisk together the soy sauce, brown sugar, minced garlic, grated ginger, beef broth, and sesame oil until the sugar dissolves. Pour the sauce over the beef and stir to coat evenly.
Cover and cook on low for 5 hours. In a small bowl, mix the cornstarch with 2 tablespoons of water to make a slurry. Stir the slurry into the slow cooker during the last 30 minutes of cooking to thicken the sauce.
Add the broccoli florets in the final 30 minutes so they soften but still hold some texture. Stir gently to combine everything with the sauce.
Serve hot over rice. Top each portion with sliced green onions and sesame seeds.