Put the flour, cake flour, salt, and cocoa in the bowl of a metal-bladed food processor and pulse briefly to mix.
Add all the water and pulse until a ball forms.
The dough will be very moist 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.

Score the top of the ball with an X to prevent the dough from shrinking, then wrap it in plastic wrap and refrigerate for 5 minutes.

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

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

Fold the rectangle in 3, folding the first third towards you and the third on top.
Refrigerate for 30 minutes.
Remove the dough from the refrigerator and turn it 1/4 turn.
Roll out the dough again into a rectangle three times as long as it is wide and fold it again, folding the first third towards you and the third on top.
Make a 1/4 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 puff pastry:
On a floured 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. Moisturize the inner contour of the full shape and place the pre-cut shape on top. Brush the top of each puff pastry with beaten egg and refrigerate for 20 minutes.

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