Slice the beef: Slice the chuck roast thin, about a quarter-inch thick, cutting against the grain so the meat cooks up tender instead of chewy.
Mix the sauce: In a bowl, whisk together the soy sauce, beef broth, brown sugar, minced garlic, and grated ginger until the sugar dissolves completely.
Layer in the crockpot: Place the sliced beef in the bottom of the crockpot and pour the sauce mixture evenly over the top.
Cook on low: Cover and cook on low for 6-7 hours, or on high for 3-4 hours, until the beef is tender and easily pulls apart with a fork.
Make the slurry: In a small bowl, whisk the cornstarch with the cold water until smooth, making sure there are no lumps.
Thicken the sauce: Stir the cornstarch slurry into the crockpot and cook on high, uncovered, for 10-15 minutes until the sauce thickens and turns glossy.
Add the broccoli: Add the broccoli florets to the crockpot, cover, and cook for 20-30 minutes until they turn bright green and tender but still slightly crisp.
Finish with sesame oil: Stir in the sesame oil for a nutty aroma and a final glossy sheen on the sauce.
Garnish and serve: Sprinkle with sliced green onions and sesame seeds, then spoon the beef and broccoli over steamed rice.