In a medium bowl, whisk together the soy sauce, beef broth, brown sugar, sesame oil, minced garlic, and grated ginger until the sugar is fully dissolved.
Place the thinly sliced beef into the slow cooker and pour the prepared sauce over the top, tossing gently to ensure all the beef is coated.
Cover the slow cooker and cook on LOW for 5-6 hours or on HIGH for 2-3 hours, until the beef is very tender.
About 30 minutes before serving, in a small bowl, stir together the cornstarch and cold water to create a smooth slurry.
Stir the cornstarch slurry into the slow cooker, then add the broccoli florets, stirring to combine. Cover and cook for an additional 30 minutes on HIGH, or until the sauce has thickened and the broccoli is tender-crisp.
Serve the beef and broccoli hot over a bed of cooked white rice.