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Line a 43 x 30-cm (17 x 12-inch) baking sheet with parchment paper.
Roll out the puff pastry into a 3-mm (1/8-inch) thick 40 x 20-cm (16 x 8-inch) rectangle. Put the dough onto the baking sheet and refrigerate for 30 minutes.
With the rack in the middle position, preheat the oven to 200 °C (400 °F).
Prick entire surface of the dough with a fork. Brush with the milk and sprinkle with the sugar. Bake for about 15 minutes. Let cool.
In a small saucepan, pour the milk. With the tip of a knife, remove the seeds from the vanilla bean and add to the milk. Gently heat the milk until it simmers.
In a saucepan, combine the sugar and flour. Add the eggs and whisk until the mixture is light and fluffy. Whisk in a little hot milk to the egg mixture, and then continue with the remaining milk. Cook over medium heat while whisking constantly.
Remove from the heat at the first boil. Transfer to a bowl and place plastic wrap directly on the hot cream. Let cool.
Refrigerate until completely chilled, about 3 hours.
In a bowl, whip the cream, sugar and vanilla. Keep cold.
In a bowl, combine the icing sugar with the water. Set aside.
Slice the baked and cooled puff pastry rectangle crosswise into 3 even strips. Cover a first dough strip with pastry cream and cover with a second strip of dough. Top with whipped cream, then cover with the last piece of puff pastry. Frost the top of the mille-feuille.
Melt the chocolate in the microwave oven. Pour the chocolate in a small cone-shaped parchment paper and cut the tip.
Draw straight lines on the frosting, then run a knife in the opposite direction to form a marbled pattern.
Serve chilled and keep refrigerated.