1Add the flours, salt, and sugar to a mixer bowl. Mix well.
2In a saucepan, heat a small amount of milk on medium and dissolve the yeast.
3Add the yeast to the bowl and turn on the mixer to medium speed using the hook attachment.
4Add the remaining milk, minus 20 grams, and continue kneading. Add the butter and knead until the dough is compact and soft.
5Place the dough in a bowl, cover it with plastic wrap and refrigerate it for one hour for the dough to rest.
6In the meantime, prepare the remaining by placing the butter on a sheet of plastic wrap, place another sheet of plastic wrap on the butter, beat on the butter in order for it to acquire a square shape approximately a few centimeters thick.
7After 40 to 50 minutes, take the dough out of the refrigerator to begin with the “flaking” procedure.
8Roll out the dough into a square shape approximately 2 centimeters thick. Place the butter in the center of the square of dough and fold the dough onto the butter wrapping it like a package.
9Once the butter is inside the dough, roll it out towards the top and the bottom (with the dough positioned in front of you).
10Fold the side of dough closest to you over, toward the center. Then, the side furthest from you from that. This will give you a three-way fold and a rectangle to be refrigerated covered in plastic wrap for 30 minutes. This step is called the "first turn."
11After the dough has been in the refrigerator for 30 minutes, position it in front of you so that the fold is always to your right and roll out the dough as before. Refrigerate the dough for another 30 minutes.
12These steps are to be repeated for at least 3 total turns.
13Finally roll out the dough into a rectangle approximately 3.4 centimeters thick, then cut it into rectangles measuring 10 centimeters by 5 centimeters.
14Place a piece of dark chocolate on the bottom of each rectangle and fold the puff pastry onto itself to seal closed.
15Once all of the rectangles have been completed, place them on a parchment paper-lined baking sheet and cover with plastic wrap.
16Refrigerate the dough for at least 5 to 6 hours. You may plan to refrigerate the dough until midnight, then let the pain au chocolat rise in the oven (the oven should be off) overnight.
17Preheat the oven to 200 degrees Celsius.
18Brush the dough with a beaten egg yolk and remaining milk and bake for 15 minutes at 200 degrees Celsius. Lower the temperature to 180 C for about 10 to 15 minutes.