Make horseradish mayo
In a small bowl stir together the mayonnaise and horseradish. Cover and refrigerate until ready to serve.
Season the steaks with salt and pepper.
Sear the steaks
Heat a large cast iron skillet over medium-high heat, add the vegetable oil and heat through. Add the steak and cook until the bottom develops a golden-brown crust, about 5 minutes.
Flip and continue to sear
Flip the steak and continue to cook it until medium-rare, about an additional 4 minutes. The internal temperature of the steak should be 130ºF (54ºC).
Let the steak rest
Remove to a cutting board and let rest for about 15 minutes. Thinly slice the steak against the grain, discarding any large pieces of fat.
Make arugula salad
In a small bowl toss the arugula with the olive oil and white wine vinegar.
Toast the ciabatta
Return the pan to medium-low heat and heat through. Add the butter and allow to melt. Add the ciabatta rolls, cut side down and toast until golden brown, about 2 minutes.
Spread the horseradish mayo over the bottom of the rolls. Divide the sliced steak over the top followed by a slice of cheese. Either use a kitchen torch to melt the cheese or heat the oven to broil on high and cook until the cheese has melted about 1 to 2 min.
Top with a handful of the arugula and close the sandwich. Cut in half and serve immediately.