Heat vegetable oil in a large skillet over medium-high heat. Pat the beef strips dry, season lightly with salt and pepper, then sear in batches until nicely browned on both sides, about 2 minutes per side. Transfer to the slow cooker.
In a medium bowl, whisk together soy sauce, brown sugar, garlic, ginger, and beef broth. Pour the mixture over the beef in the slow cooker. Stir gently to coat.
Cover and cook on HIGH for 4 hours or on LOW for 6–7 hours, until the beef is fork-tender and reaches an internal temperature of 145°F (63°C). Resist the urge to lift the lid; it adds 15–20 minutes cooking time each peek.
After initial cook time, whisk together cornstarch and cold water to make a slurry. Stir into the slow cooker. Add broccoli florets, cover, and cook on HIGH for an additional 30 minutes, or until broccoli is crisp-tender and sauce has thickened. Stir in sesame oil just before serving.
Taste and adjust seasoning with a little more soy sauce if needed. Serve the beef and broccoli over fluffy jasmine rice, showered with sesame seeds and scallions.