Heat a large stainless steel or cast-iron skillet or griddle over medium-high heat. Sprinkle one side of the steak with 1/2 teaspoon salt and 1/4 teaspoon pepper. Add 2 tablespoons of the oil to the hot pan and heat. Add the steak seasoned-side down and sprinkle the other side with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook until the steak reaches an internal temperature of 130 to 135 degrees F, about 7 minutes per side; the steak will keep cooking as it rests. Let it rest for 7 to 10 minutes.
While the steak is resting, add the mint, parsley, capers, vinegar and anchovies to a food processor (or chop and add to a bowl). Pulse until finely chopped. Add the lemon juice and remaining oil and pulse until combined. Season with salt and pepper. Put the salsa into a bowl.
Slice steak into thin strips, about 1/4-inch thick.
Layer the grated cheese on the cut sides of the bread loaves. Broil until melted and bubbly.
Evenly layer the steak, salsa verde and arugula on the bread. Serve immediately.