Cream the butter, sugar and half of the cinnamon together.
Peel then grate the apples using a coarse cheese grater. Add the cinnamon, lemon juice and zest, then combine.
Grind or finely chop the walnuts, if used.
Roll out the puff pastry, then spread the cinnamon butter evenly on top. Scatter the apple mixture on top then sprinkle with the ground nuts.
Roll up the pastry along its long side.
Cover the roll tightly with cling-film and chill for 15 minutes. Meanwhile, preheat the oven to 200°C.
Take the roll out of the fridge and cut it into 1.5 cm slices using a sharp knife. Place these on a lined baking sheet, allowing plenty of space between them (they will double in size).
Sprinkle them with a little sugar, if you prefer a crunchy finish, or brush them with an egg-wash for a golden glazed finish.
Bake for 12-15 minutes until golden, then transfer them onto a wire rack to cool.