Preheat the broiler.
In a large skillet, heat about 3 tablespoons olive oil, 3 turns of the pan, over medium-high. When the oil ripples, add the shallots, chopped garlic, chile and oregano. Stir until the shallots soften, about 2 minutes. Add the wine; bring to a simmer. Add the tomatoes, season with salt and pepper. Cook the sauce until the tomatoes are soft and have released their juices, 15-20 minutes. Add the parsley and remove from the heat.
On a baking sheet, arrange the rolls, cut-side up. Broil until golden brown, about 2 minutes. (Keep the broiler on.) Rub the cut sides of the rolls with the halved garlic, drizzle with olive oil and season with sea salt. Remove the roll tops from the baking sheet.
Season the steaks. When the tomatoes have been cooking about 15 minutes, heat about 1 tablespoon olive oil, one turn of the pan, in a cast-iron or other heavy skillet over medium-high. When the oil ripples, add the butter. When the butter foams, add the steaks. Cook until browned and caramelized, about 3 minutes per side. Transfer steaks immediately to the roll bottoms and top each steak with 2 slices mozzarella. Broil until cheese is melted and the edges are browned, 2-3 minutes.
Top the sandwiches with some tomato sauce, a mound of arugula and set the roll tops in place.