Season the ribeye steak with salt and pepper. Heat a cast-iron skillet over medium-high heat, add oil, and sear the steak for 4-5 minutes on each side for medium-rare. Let it rest afterward.

In another pan, melt butter over medium heat and sauté sliced onions until caramelized (about 15-20 minutes).

Slice rolls in half and lightly toast them in the skillet used for the steak.

Place cheese on toasted rolls and cover briefly until melted.

Assemble sandwiches by layering sliced steak, caramelized onions, fresh arugula, and horseradish sauce on each roll.

Serve immediately while warm.