Preheat the oven to 450 degrees. In a large skillet, melt the butter. Add the apples and cook over moderate heat, stirring occasionally, until barely softened and just beginning to brown, about 5 minutes. Add the brown sugar and cook over moderately low heat, stirring occasionally, until the apples are tender and lightly caramelized, about 10 minutes longer. Stir in the cinnamon and keep warm.

Heat a grill pan. Grill the hot dogs over high heat until lightly charred all over, about 5 minutes. On a baking sheet, set the dogs in the rolls and top with the cheddar cheese. Bake for about 3 minutes, just until the cheese is melted. Top with the apples and serve at once.

The apples can be refrigerated overnight. Rewarm before serving.