Preheat the oven to 190C/375F/Gas 5.
For the millefeuilles, place the puff pastry rectangles onto a baking sheet and place another baking sheet on top (this is to to keep the puff pastry flat as it cooks). Transfer to the oven and bake for 10-15 minutes, or until the pastry has risen and is golden-brown. Remove from the oven, remove the uppermost baking tray and set aside to cool.
Sprinkle two and a half tablespoons of the icing sugar onto a plate. Dredge the pear slices in the icing sugar. Heat a griddle pan over a medium heat until hot, then cook the sugared pear slices for 1-2 minutes on each side, or until golden-brown griddle marks appear on the pears. (You may need to do this in batches.)
Place the double cream, rum and the remaining half tablespoon of icing sugar into a bowl and whisk until firm peaks form when the whisk is removed.
Whisk in the chestnut purée, then fold in the chopped chestnuts.
For the caramel, gently heat the sugar in a non-stick pan until the sugar is dissolved and has caramelised. Continue to cook for 4-5 minutes, or until the mixture is dark golden-brown. Remove from the heat and whisk in half of the double cream.
In a separate bowl, whisk the remaining double cream until firm peaks form when the whisk is removed.
Pour the caramel mixture onto the whipped cream and fold together.
To assemble the millefeuilles, spread a thin layer of chestnut cream onto four of the puff pastry rectangles. Top each with a few pear slices.
Place another piece of puff pastry on top of each millefeuille. Spread another layer of chestnut cream onto the puff pastry, then top with a few more slices of pear. Finish each millefeuille with a third puff pastry rectangle. Dust with icing sugar.
To serve, spoon the caramel cream sauce into the centre of four serving plates and place one millefeuille into the centre of each plate.