Dissolve the yeast in the warm water, and let sit for 5 minutes.

In a stand mixer fitted with a paddle attachment, mix the flour, salt, and softened butter on medium-low until just combined, about 2 minutes.

Add the yeast mixture to the flour. Change out the paddle attachment to a dough hook. Mix until the dough begins to come together. Add the cold water and mix until combined.

Add the cold butter to the flour mixture. Mix on medium speed just until the dough forms a ball, leaving large butter chunks in the dough, about 3 minutes.

Dust a flat surface with flour and turn out the dough onto surface. Using a rolling pin, roll out into a 12x15-inch rectangle. Dust with small amounts of additional flour as needed to prevent sticking.

Fold the bottom half of the dough up to center, then the top half over the bottom.

Starting from the left side, roll the dough to the right side until it forms one large roll. Press the dough down with your hand to roughly 2-inches thick. Cover with plastic wrap and rest dough at room temperature for 30 minutes.

Press down the dough again. Wrap fully with plastic wrap, and transfer to the refrigerator. Chill for 1 hour.

Spray a 12-cavity muffin pan with nonstick cooking spray.

Sprinkle half of the sugar over a clean surface. Place the chilled dough on top of the sugar, and sprinkle remaining sugar over top of dough.

Roll the dough out to a 16 inch square.

Fold 1/3 of the dough from the right side to center. Fold the left side over the other two layers of dough.

Roll out the dough to a 13x17-inch rectangle.

Using a sharp knife, trim off 1/2 inch of dough on all sides. Discard scraps.

Using a ruler, cut dough into twelve 4 inch squares.

Pinch the top corners of each square together, and place in the cavities of the muffin pan.

Cover the muffin pan loosely with plastic wrap and proof at room temperature for 45 minutes.

Preheat the oven to 400°F.

Bake the kouign-amann on middle rack of oven for 5 minutes.

Reduce the oven temperature to 350°F and continue baking until the pastry has browned all over, about 40 minutes.

Check to see that kouign-amann are done. Remove from oven or add time as needed.

Immediately remove the kouign-amann from muffin pan with a fork or butter knife and transfer to a cooling rack.

Cool at room temperature for 5-10 minutes before serving. Leftovers can be stored in an airtight container for up to 3 days.