In a large bowl, stir together the milk and yeast. Stir in the sugar, eggs and vanilla and mix well. Add a cup of the flour and the salt, then add the rest of the flour gradually, stirring until it's incorporated. Knead the dough on a lightly floured countertop for about 5 minutes, until smooth. Transfer to a lightly floured baking sheet and cover with plastic wrap; chill in the refrigerator for 30 minutes.
Meanwhile, beat the butter and flour with an electric mixer for a couple minutes, scraping down the sides of the bowl, until smooth. Set aside (don't refrigerate).
When the dough has chilled, turn it out onto a lightly floured surface and roll into a rectangle that is about 13"x18" and 1/4" thick. Spread the butter evenly over the right two-thirds of the dough. Fold the left third of the dough over, covering half the butter, then fold the right side over, as if you were folding a letter in thirds. (Unlike a letter, the dough ends should line up, so that it's folded in three.) Cover the dough in plastic wrap and put it back into the fridge for 30 minutes.
Put the dough back on the floured surface lengthwise, with the open sides to the left and right. Roll it out into another 13"x18" rectangle, 1/4" thick. Fold the left third over the middle, then the right third over the middle. (This is referred to as "turns". To keep track of each fold -or turn- press your finger into the dough at the edge to make two marks - you can do this each time you roll and fold so that you know how many times you've done it.) Chill the dough for another 30 minutes.
Roll, fold and refrigerate the dough two more times, so that you've done it four times total. Cover and refrigerate for at least 5 hours, or overnight. It can also be frozen at this point for up to 4 months.
To assemble the pain au chocolat, take the dough out of the fridge and roll it on a lightly floured surface to about 1/4" thick. You can cut the dough into rectangles as large or as small as you like - we made them on the small side, cutting the dough into strips and then crosswise so that each piece was about the size of a business card. Put a little pile of chocolate, or a chunk of it, along the middle of the pastry, roll the sides up and place each one seam-side down on an ungreased baking sheet. If you have time, cover loosely with plastic or a tea towel and let them proof for an hour or two. (This is not absolutely necessary- we did ours in a rush!)
Preheat the oven to 400F. Bake the pain au chocolat for 15 minutes, until golden. (If they are larger, they may need more time.) Try to be a good person and share with your family and friends.