Kouign Amann are buttery layers of bread dough folded with a generous sprinkling of sugar in between the thin pastry layers.
In a small bowl, combine yeast, a pinch of sugar and warm milk. Stir and let stand for 10-15 minutes until thick and foamy. Combine flour and salt in the bowl of your stand mixer. Add to it the yeast mixture. With the dough hook attachment, slowly mix the ingredients until moistened. Knead the dough on low speed for 4-5 minutes until it is smooth, soft and cleaning the sides of the bowl. Gather the dough into a ball and place it inside a bowl. Let it rise for an hour, covered with plastic wrap. After rising, gently deflate the dough and pat it into a rough rectangle. Wrap it in plastic and chill for 40 minutes.
Add about 3 teaspoons of flour inside a 7 x 8 inches Ziploc bag with seal. Shake it around to distribute. Add the cubed butter inside the bag and spread them in a single layer. Using your hands and fingers, smash the butter quickly so they become a little flexible for shaping. Once they are pliable, roll the rolling pin over the butter until the butter forms a rectangle and has reached all the corners of the bag. It is essentially a rectangle that is the same size as your Ziploc bag. Chill in the fridge until needed.
In a floured surface roll the dough into a rectangle that is 17 x 10 inches in size. It should be about 1/4 inch thick. Take the butter out from the fridge, use scissors to rip Ziploc open and peel it off from the butter. Position the butter at the center of the rectangle dough. The long side of the butter should be parallel to the short side of dough. Fold the lower third of the dough over to the center completely covering the butter. Fold the upper third of the dough towards the center to cover the first fold. You have done a business letter type of fold.
Rotate the dough so that the short side is facing you. Flour the board and your pin as needed. Roll the dough into 17 x 10 inch rectangle, then do the business letter fold again: Fold the lower third to the center, then upper third over it. That is your first turn. Repeat the process one more time for your second turn, ending with the business letter fold again. Wrap in plastic and chill in the fridge for 30 minutes.
Take dough out from the fridge and with the short side facing you, roll it out again to a 17 x 10 inches rectangle. Sprinkle 1/2 cup of sugar evenly across the entire rectangle surface. Roll the rolling pin once over the sprinkled sugar to press them deep into the dough. Fold the lower third over the center and fold the upper third of dough over that. This is your third turn. Rotate the dough again so that the short side is facing you. Roll again to 17 x 10 inches rectangle. Sprinkle 1/2 cup of sugar over the dough surface, roll a rolling pin over it to press then fold the dough, business letter style. This is your fourth and final turn. Chill dough in the fridge for 30 minutes, wrapped in plastic.
Sprinkle 1/4 cup of sugar on your working surface. Roll the dough to an 8 x 24 inch rectangle. Using a knife or pizza cutter, cut it down vertically in half. Cut 6 equal sections horizontally to form 12 squares that are about 4 inches in size.
Grease a muffin pan that has 12 holes. Take each square pastry, fold the corners to the center and place each folded dough on a muffin hole. Cover the pastries loosely with plastic wrap and let rise for 40 minutes.
Pre-heat oven to 400 F and maintain temperature for about 15-20 minutes. Set the muffin pan on a baking sheet that is lined with foil to catch any butter drips. Put the pastries in the oven and immediately lower temperature to 350 F. Bake for 35-40 minutes. Rotate the pan halfway into the baking. Let pastries cool slightly then transfer to a cooling rack. Let the pastries sit for a few minutes to allow the dough inside to set before consuming. They will be doughy and moist initially, but will set into the right texture as they sit.
This recipe is a combination of the methods and principles I learned from The Kitchn, and the book World Class Cakes by Roger Pizey.