Bring a large pot of water to a boil over high heat. Add in broccoli florets, submerge, and boil for 90 seconds. Remove from the water, strain, and rinse with cold water to stop the cooking. Set aside.
In a sauté pan over medium heat, heat olive and sesame oil. and add oils. Add garlic and cook for 30 seconds. Stir in the soy sauce and ginger and simmer for 5 minutes.
Add in the brown sugar and continue cooking another 5 minutes.
Increase heat to medium high, add the shrimp, and cook just until pink, about 1 minute per side. Stir in the green onions and the broccoli. Cook another 2-3 minutes.