Whisk together brown sugar, Sriracha, and egg whites in a shallow dish. Combine panko and sesame seeds in a shallow dish. Working in batches, add shrimp to egg mixture; dredge in panko mixture.
Heat 1 1/2 teaspoons canola oil and 1 1/2 teaspoons sesame oil in a large nonstick skillet over medium-high. Add half of dredged shrimp; cook until golden brown, 2 to 3 minutes per side. Remove from pan. Repeat procedure with 1 1/2 teaspoons canola oil, remaining 1 1/2 teaspoons sesame oil, and remaining half of dredged shrimp. Sprinkle cooked shrimp with 1/4 teaspoon salt and 1/4 teaspoon pepper.
Heat remaining 1 tablespoon canola oil in pan over medium-high. Add garlic and ginger; cook, stirring constantly, 1 minute. Add broccoli; cook until lightly browned, about 2 minutes. Add 1/4 cup water; cover and cook until crisp-tender, 2 to 3 minutes. Uncover; sprinkle with remaining 1/4 teaspoon salt and remaining 1/4 teaspoon pepper. Serve with shrimp.