In your mixing bowl add milk, yeast and sugar together and let proof about 10 mins until the yeast is dissolved.
Add the flour, salt and 2 Tbsp softened butter to the warm milk mixture. Mix with a dough hook until it's combined, then mix 2-3 more mix until it's smooth (if you don't have a bosch you can knead by hand about 5 mins)
Let the dough sit (covered) for 1 hour (or until doubled in size.
With your hands, on the counter (or in your Bosch) mix the cold butter and 3 Tbsp flour until it is smooth and without lumps. Work quickly to keep the butter cold.
Place the butter on plastic wrap and shape it into a 9x6 in rectangle. Wrap and refrigerate the butter 30 mins
knock the dough down, knead lightly in the bowl and refrigerate for 30-45 mins.
Roll out the dough onto a lightly floured surface to a 15 x 8 inch rectangle.
Take the rectangle of butter and place over 2/3 of the dough leaving a 1/2 inch border on 3 of the side and about 5 inches on the other. Fold over the excess 5 inches of dough then fold over the buttered side like you were folding a business letter. Seal the edges and rotate 1/4 turn.
Roll out the dough/butter into a 18x8 inch rectangle, fold into thirds again, wrap in plastic wrap and refrigerate 30 mins (first turn)
Repeat rolling out, folding, wrapping and refrigerating 2 more times (2nd and 3rd turns)
Roll out the dough one more time, roll into thirds again, wrap in plastic wrap and refrigerate, this time for an hour or overnight (4th turn) (at this point the dough can also be frozen, let thaw completely before proceeding.).
Roll out dough into a rectangle about 10 x 15 inches (it will be about 1/4 inch thick)
Cut down the center, creating two 5 x 15 inch rectangles
Cut small rectangles by cutting every 3 inches, creating 12 5x3 inch rectangles
Place one of the chocolate stick at one of the 3 inch ends, roll it up until it's completely wrapped and press the dough together
Add a second stick just on the other side wrap around the rest of the dough, press the seam so it's underneath
Cover with plastic wrap sprayed with non stick spray, like pam or bakers joy, and let rise until increased by almost half, 1 1/2 - 2 hours.
Brush each pain au chocolat lightly with the egg. Bake at 375 for 20-25 mins.