Whisk the marinade ingredients. Place the steak in a large, resealable plastic bag and pour in two-thirds of the marinade (about ½ cup). Press the air out of the bag and seal tightly. Turn the bag to distribute the marinade and place the bag in a bowl. In a separate bag combine the remaining marinade (about ¼ cup) and the scallions. Refrigerate the steak and scallions for at least 2 hours or up to 4 hours, turning the bags occasionally. Allow the steak to stand at room temperature for 15 to 30 minutes before grilling.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Whisk the dressing ingredients, except the oil, and then slowly add the oil until the dressing is emulsified.
Remove the steak and scallions from the bags and discard the marinade. Grill the steak over direct medium heat, with the lid closed, until cooked to your desired doneness, 7 to 9 minutes for medium rare, turning once or twice. At the same time, grill the scallions over direct medium heat until lightly charred and slightly softened, 3 to 5 minutes, turning once. Remove from the grill and let the steak rest for 5 minutes. Cut the scallions crosswise into 1-inch pieces. Cut the steak across the grain into thin slices.
In a large bowl toss the arugula with 2 tablespoons of the dressing. Divide the arugula and steak slices among four plates. Top with more dressing and the scallions. Serve warm.