In a small bowl, mix together garlic, ginger, honey, and soy sauce.
Place raw shrimp in a large zip lock bag. Pour 1/3 of the honey soy sauce in the bag, then massage the bag to coat shrimp. Seal bag and let shrimp marinate in the refrigerator for 1 hour. Keep the remaining honey soy sauce covered and at room temperature.
Warm a large skillet over medium high heat. Add 2 tablespoons of reserved honey soy sauce to pan and let heat until steaming. Toss broccoli in skillet and cook until warm and tender, about 5-8 minutes. Remove broccoli from skillet and set aside.
Remove shrimp from zip lock bag and discard the bag and marinade. Place 5-10 shrimp in the skillet at a time and cook for 1 minute on each side or until shrimp bodies are curled and browned. Before removing from pan, use tongs to rub shrimp into the caramelized sauce on the bottom of the skillet.
When the shrimp are cooked, return all shrimp and broccoli to skillet. Heat for 2-3 minutes, tossing to catch any remaining caramelized sauce on the bottom of the skillet.
Serve immediately with remaining honey soy sauce drizzled on top.