Preheat oven to 375°F. Grease an 8- or 9-inch round baking pan; set aside. Unroll dough (do not separate); press perforations to seal. Brush dough with melted butter. In a small bowl, stir together the granulated sugar and cinnamon; sprinkle over dough. Starting from a long side, roll up dough. Using a sharp knife, slice dough into eight pieces. Arrange rolls, cut sides up, in prepared pan, flattening each slightly.
Bake for 15 to 18 minutes or until golden. Cool rolls slightly in pan on a wire rack. Remove rolls from pan. In a small bowl, stir together powdered sugar, vanilla, and enough orange juice or milk to make an icing of drizzling consistency. Drizzle icing over warm rolls. Serve warm.