In a small bowl, mix together the softened butter, brown sugar, and cinnamon until you have a smooth, spreadable paste.
Unroll the sheet of puff pastry onto a lightly floured surface, keeping the parchment paper underneath for ease of handling.
Evenly spread the cinnamon-sugar mixture over the entire surface of the puff pastry, making sure to reach the edges.
Starting from one long side, tightly roll up the pastry into a log. Use the parchment paper to help guide the roll if needed.
Preheat your oven to 200°C (180°C fan) and line a baking tray with parchment paper.
Remove the chilled roll from the refrigerator and cut it into slices about 1 cm thick.
Place the slices flat on the prepared baking tray, spacing them slightly apart to allow for puffing during baking.
Brush the tops of the swirls with the beaten egg for a golden finish.
Bake in the preheated oven for 12-15 minutes, or until the swirls are golden brown and puffed.
Optional Toppings:
Let the swirls cool slightly, then drizzle with icing sugar glaze (made by mixing icing sugar with a small amount of water or milk) or dust with extra cinnamon sugar for added sweetness.
Serve warm or at room temperature. These swirls are perfect with a cup of coffee or tea!