In a small mixing bowl, whisk together the horseradish, mayonnaise, sour cream, and Parmigiano Reggiano. Cover and place in the refrigerator.
Ribeye
Season both sides of each ribeye steak liberally with the rub.
Light the lump charcoal in a chimney, once glowing, carefully transfer the coals to the grill.
Place the seasoned ribeye steaks on the grill. Cook for 5 to 7 minutes per side, depending on thickness and doneness preferred. Remove from the grill and let rest for 7 to 10 minutes.
Sandwich
Brush the sliced ciabatta rolls with olive oil, and place them on the grill cut-side-down. Remove the rolls once they are warm with a slight crunch to the cut side.
Slice the steaks thinly on the bias (across the grain).
Slather the top and bottom of each roll with the premade horseradish dressing. Lay thin slices of red onions on the bottom of each roll. Add a generous pile of sliced ribeye steak. Add a handful of fresh arugula to the top of the steak, and close with the top of the roll.