In a small bowl, whisk together soy sauce, cornstarch, water, brown sugar, and sesame oil until smooth. Set aside.

Heat vegetable oil in a large skillet over medium-high heat. Add shrimp and cook for about 2-3 minutes per side, until pink and opaque. Remove shrimp from skillet and set aside.

In the same skillet, add broccoli and more oil if needed. Cook for about 2 minutes. Then, add the garlic, ginger, bell peppers, and carrots.

Stir-fry for about 3-4 minutes until vegetables are just tender.

Return the shrimp to the skillet. Pour the prepared sauce over the shrimp and vegetables. Cook for an additional 1-2 minutes, stirring constantly, until the sauce has thickened and everything is heated through.

Serve immediately over rice, sprinkled with green onions.