- 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.