Preheat a grill to medium-high heat. In a medium skillet, heat 2 tablespoons olive oil over medium heat. Add the onions,season with salt and pepper and cook until golden-brown, about 15 minutes.
Brush the steak with the remaining 1 tablespoon olive oil and season with salt and pepper. Grill, turning once, for about 12 minutes for medium-rare; let rest for 10 minutes before thinly slicing.
Meanwhile, in a small bowl, stir together the mayonnaise, horseradish and a pinch each salt and pepper. Spread onto one side of each bread slice.
Divide the sliced beef among 4 bread slices, then top with the caramelized onions and the arugula and set the remaining bread slices into place. Serve with the celery sticks.