Let's start with the pastry cream. Heat the milk with the vanilla seeds and the open vanilla pod. Leave to infuse for at least half an hour.
In a bowl, whisk together the egg yolks and sugar. Then add the cornstarch and the sifted flour. Mix well, then mix with the vanilla milk, taking care to remove the vanilla pod.
Return everything to the saucepan and heat, stirring constantly, until the cream thickens. Not too much especially, because it will thicken even more when it cools. Pour into a container and add the butter. Mix and film on contact. Leave to cool while you prepare the rest.

For the almond cream, mix the eggs, ground almonds, sugar and butter. The whole must be homogeneous. Pour the pastry cream into the almond cream and add the rum. Mix well.

Preheat your oven to 200 ° C 180 ° C.
 

Roll out a puff pastry and distribute the frangipane over it, leaving 3 cm of free edge. Don't forget the bean !!

Using a food brush , wet the edges of the dough with water. Cover with the second puff pastry and press the edges to seal them.
Drill a hole in the center of the cake to let the steam escape. With the blade of a knife, trace the patterns of your choice, then brush with egg yolk.

Bake for 45 minutes. Just before removing from the oven, heat the water and sugar together for the syrup. As soon as the galette is cooked, brush it with syrup with a food brush .

Let the galette cool or at least cool down before eating. You can also eat it later and reheat it for 10 minutes in the oven at 160 ° C.