Our version of the classic sandwich becomes a hearty meal perfect for lunch or a light dinner.
In a large baking dish, whisk together mustard, 1 tablespoon vinegar, and 2 tablespoons oil. Season with salt and pepper. Add steak and turn to coat; let sit 15 minutes at room temperature. Heat a grill pan or cast-iron skillet over medium-high. Drizzle onion with 1 teaspoon oil and season with salt and pepper. Grill steak and onions until steak is medium-rare and onions are tender,5 to 7 minutes, flipping once. Let steak rest 5 minutes before thinly slicing against the grain. Meanwhile, in a food processor, process onions until finely chopped, then mix with 1/2 teaspoon vinegar. Season to taste with salt and pepper. Spread onion mixture on 4 slices bread. Top with steak, arugula, Parmesan, and 4 more slices bread.