To prepare the dough:
Heat the milk, water, sugar, salt, and butter in a saucepan until the butter is completely melted.
Remove the saucepan from the heat and add all the flour, stirring vigorously until a dough ball forms.
Return the saucepan to the heat and simmer for a few minutes, stirring constantly, until the mixture pulls away from the sides of the pan.
Let it cool slightly, then add the eggs one at a time, stirring well after each addition, until a smooth, even dough forms.
To bake the dough:
Preheat the oven to 230°C (450°F).
Spread the dough onto a baking sheet (45 x 35 cm / 18 x 14 inches) lined with parchment paper and smooth it with a spatula.
Bake for 10 minutes at 230°C (450°F), then reduce the temperature to 200°C (400°F) and bake for another 10 minutes, or until golden brown.
Let it cool completely.
To prepare the filling: Heat the milk with the vanilla bean (halved and scraped) in a saucepan.
Mix the vanilla powder, egg yolks, and sugar in a bowl.
While whisking constantly, add the egg yolk mixture to the hot milk until the vanilla cream thickens.
Remove the pot from the heat and let it cool completely.
Cream the room-temperature butter until fluffy and add it to the cooled vanilla cream. Stir until smooth.
To assemble the cake:
Cut the cooled cake horizontally in half.
Spread a layer of cream on the bottom half of the cake and top with half of the sliced strawberries.
Place the other half of the cake on top and spread another layer of cream over it.
Decorate with the remaining strawberries and dust with powdered sugar.
