Step 1Stir together the one-third cup balsamic vinegar, 3 tablespoons of olive oil and the rosemary. Place the steaks in a sealable plastic bag and pour in the liquid mixture. Squeeze out the air, seal tightly and move the steaks around to make sure they are all well-coated with the marinade. Refrigerate for at least an hour or up to overnight.
Step 2Heat a grill over medium-high heat. Meanwhile, make a vinaigrette by whisking together the lemon juice, Champagne vinegar and shallots, then whisking in the three-fourths cup olive oil. Season with 1 1/4 teaspoons salt and pepper to taste. Taste and add more salt if necessary, then set aside.
Step 3Season the steaks generously with salt and pepper. Cook until well-seared on both sides and medium rare, 4 to 5 minutes per side. Transfer the steaks to a plate and let rest for 3 to 4 minutes. Using a sharp knife, carve the steak diagonally into 1-inch strips across the grain.
Step 4Rinse and dry the arugula and place it in a large bowl. Whisk the vinaigrette briefly, then pour it over the arugula and toss gently to coat lightly. Divide the sliced steaks among 6 to 8 dinner plates and place a mound of salad beside each. Using a vegetable peeler, shave a few shards of Parmigiano-Reggiano over the arugula. Drizzle the steak lightly with aged balsamic vinegar and olive oil and serve.