In nonstick 12 inch skillet, heat 2 teaspoons oil over medium high heat.
Add green onions and mushrooms and cook until tender and brown, about 5 minutes, stirring often.
Remove to bowl.
In same skillet, heat 1 teaspoon oil.
Add half the beef and cook, stirring constantly, until beef just loses its pink color.
Remove to bowl with vegetables.
Cook remaining beef as above, adding remaining 1 teaspoon oil.
In cup, mix soy sauce, balsamic vinegar, and brown sugar.
Return beef mixture to skillet; stir in soy sauce mixture.
Cook 1 minute to heat through, stirring.
Remove from heat; stir in half the arugula.
Spoon beef mixture over remaining arugula on platter.