In a small pan, heat 1 tablespoon of butter. Sauté onions 8-10 minutes or until tender and slightly caramelized. Set aside.
Heat a nonstick or cast-iron pan over medium heat. Spread each slice of bread with horseradish sauce. Place cheese on bread, followed by onions, roast beef and remaining slice of bread. Spread top of bread with softened butter and half of the shredded Parmesan. Place buttered side down in heated pan. While sandwich is cooking, spread remaining softened butter and Parmesan cheese on top of bread. Cook approximately 3-4 minutes or until lightly browned. If necessary, turn heat down so the bread doesn’t burn. Flip sandwich over and cook another 3-4 minutes or until lightly browned.