Put the flour, cake flour, and salt in the bowl of a food processor equipped with a metal blade and pulse a few times to mix.
Add all the water and pulse until a ball forms.
The dough will be very wet and soft.

Remove the dough from the machine, gather it with your hands, and work it lightly, adding a few drops of water as needed, to form a soft and smooth ball.

Cut a cross on the top of the ball to prevent the dough from shrinking, then wrap it in plastic wrap and refrigerate for 5 minutes.

Roll out the dough on a floured work surface into a large square.

Spread the butter to a thickness of 1/2″ (1 cm) in the middle and fold the 4 corners over the middle. Cover and refrigerate for 30 minutes.
Roll out the dough on the floured work surface into a rectangle of 5″ x 14″ (12.5 cm x 35 cm).

Fold the rectangle into 3, folding the first third towards you and the third on top.
Refrigerate for 30 minutes.
Take the dough out of the refrigerator and turn it a quarter turn.
Roll out the dough again into a rectangle three times as long as it is wide and fold again the first third towards you and the third on top.
Make a quarter turn, mark the corner with your finger to remember, and refrigerate for 30 minutes.
Repeat the operation 4 more times. The dough will then be ready to use.

Preparation of vol-au-vents:
On a floured work surface, roll out the dough to a thickness of 1/4″ (0.5 cm). Cut out the desired shapes in pairs using a cookie cutter. Pre-cut one of the shapes in the middle for the hats. Moisten the inner edge of the full shape and place the pre-cut shape on top. Brush the top of each vol-au-vent with beaten egg and refrigerate for 20 minutes.

Baking:
Preheat the oven to 400 F (200 C). Bake the vol-au-vents for 15 minutes, then reduce the oven to 350 F (180 C) and continue baking for 15 minutes.