Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the onion and 1/4 teaspoon salt; cook, stirring, until golden brown, about 10 minutes. Transfer to a bowl; reserve the skillet.
Meanwhile, make the dip: Bring the beef broth and nutmeg to a simmer in a small saucepan; remove from the heat and add the horseradish and 1/2 tablespoon butter. Cover and keep warm.
Layer half of the cheese, then the roast beef, onion and the remaining cheese on the rolls. Melt 1 tablespoon butter in the skillet over medium heat. Add 2 sandwiches and top with a heavy skillet. Cook, pressing down on the top skillet to flatten the sandwiches, until the bread is golden brown and the cheese melts, about 4 minutes per side. Repeat with the remaining 1 tablespoon butter and 2 sandwiches. Serve with the dip.