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Heat the oven to 200C/180C fan/gas 6 and line large baking tray with baking parchment. Stir the cinnamon and sugar together in a small bowl. Unravel the puff pastry on a lightly floured work surface, then sprinkle the cinnamon sugar mixture all over the top. Gently level the cinnamon sugar mixture with the back of a spoon so it covers the pastry almost completely, leaving a 1cm border on one of the short sides. Brush a little of the beaten egg over the exposed pastry border.
Roll the pastry up in a tight log from the short side without the border. Gently press along the egg-washed border to seal, then slice into 12 equal rounds. Arrange the cinnamon rolls on the prepared baking tray, placing them up against each other so they're just touching. Brush the remaining beaten egg all over the tops and sides of the pastry, then bake for 15-18 mins until golden and risen. Leave to cool on the tray for 10-15 mins.
Combine the icing sugar with 1½ tsp water in a small bowl until you have a thick icing that easily runs off the spoon. Drizzle this over the warm rolls, then immediately serve.