Lightly brush the steak on both sides with oil and season evenly with salt and pepper. Allow the steak to stand at room temperature for 15 to 30 minutes before grilling.
Prepare the grill for direct cooking over high heat (450° to 550°F).
In a small saucepan over medium-high heat, combine the vinegar and sugar. Allow the mixture to reduce by half, 6 to 8 minutes, stirring occasionally. Remove from the heat and let cool.
Brush the cooking grates clean. Grill the steak over direct high heat, with the lid closed as much as possible, until cooked to your desired doneness, 6 to 8 minutes for medium rare, turning once or twice (if flare-ups occur, move the steak temporarily over indirect heat). Transfer to a cutting board and let rest for 3 to 5 minutes.
Cut the steak in half lengthwise and then cut each half across the grain into thin slices; divide evenly on serving plates. Pour any juices remaining on the cutting board over the steak, and pile the arugula on top. Drizzle each serving of arugula with oil and the balsamic reduction, season with salt and pepper, and top with the cheese.