1. Freeze steak 30 minutes. Cut steak across grain into 1/4-inch-thick strips.
2. Combine brown sugar and next 5 ingredients (through chile paste) in a small bowl, stirring with a whisk.
3. Place a wok or large frying pan over high heat until very hot. Drizzle oil around sides of wok to coat sides and bottom. When oil is very hot, but not smoking, add garlic; stir-fry 30 seconds or until golden. Remove garlic using slotted spoon; drain on paper towels. Add ginger, onion, and broccoli to pan; stir-fry 2 minutes. Add steak; stir-fry 2 minutes or just until no longer pink.
4. Stir in soy sauce mixture. Cook 1 minute or until sauce thickens, stirring constantly. Serve over rice. Top with garlic slices.