Combine lemon juice, soy sauce, sesame oil and 2 tablespoons of the orange juice in a resealable plastic bag. Add grouper; seal bag and refrigerate 15 minutes.
In a small nonstick skillet, heat canola oil over medium heat. Add ginger, stirring, for 30 seconds. Then add carrot and saute another 5 minutes. Add arugula or spinach, and saute until wilted. Add remaining orange juice and turn off heat.
Meanwhile, broil or grill the marinated grouper 6 inches from heat for about 8 minutes, until opaque all the way through. Serve over greens.