Put the steak in a glass or ceramic dish and add the marinade, turning the steak several times to coat. Cover and refrigerate for 2 to 4 hours.
Prepare a charcoal or gas grill. Lightly spray the grill rack with vegetable oil cooking spray. The coals should be moderately hot to hot.
Lift the steak from the marinade, letting most of the marinade drip back into the dish. Grill the meat for 8 to 10 minutes, brushing several times with the marinade during the first 5 minutes of grilling. (The oil in the marinade may cause flare-ups.) Turn and grill for 8 to 10 minutes longer until medium-rare, or until it reaches the desired degree of doneness. Let cool.
While the steak grills, lay the mushrooms and scallions near the outside of the grill away from the most intense heat, brush with oil, and grill for 8 to 10 minutes, turning once or twice and brushing with more oil, until the mushrooms are tender and the scallions are lightly browned and tender.
Slice the steak and mushrooms into strips and transfer to a bowl. Cut the scallions into short lengths and transfer to the bowl.
To make the vinaigrette, whisk together the vinegar and mustard. Still whisking, slowly add the oil, whisking until emulsified. Season with salt and pepper. Pour half the vinaigrette over the warm steak and vegetables and toss gently to coat. Set aside for about 30 minutes to cool to room temperature. (If setting aside for any longer, cover and refrigerate; bring to room temperature before serving.)
Arrange the lettuce and arugula on a platter. Spoon the meat and vegetables over the lettuce. Garnish with the tomatoes and olives and sprinkle with the cheese. Pass the remaining vinaigrette on the side, whisking well before serving.
For marinade: Whisk together all ingredients in a glass or ceramic bowl until blended. Adjust the seasoning as needed. Use according to the recipe, or cover and refrigerate for as long as 2 days.