The kouign-amann is baked in a muffin tin so the outside of each one gets deeply browned and caramelized.

Grease a standard 12-cup muffin pan with butter and sprinkle with sugar.

Sprinkle a generous amount of sugar over a clean working surface. Set the dough on top of the sugar with the shorter side facing you. Top the dough with a generous sprinkle of sugar and repeat the letter fold as described in the laminated dough process. Cut the dough into 12, 4-inch squares. Working with one square at a time, bring together its corners and set the dough into the muffin pan, with the corners facing up. Repeat with remaining 11 pieces of dough. Sprinkle the tops of the dough with sugar, cover the pan loosely, and set aside until the dough rises enough to fill each of the muffin cups, about 30 minutes.

Bake the pastries: Preheat the oven to 375°F with a rack in the center. Bake until the pastries are golden brown, about 25 minutes. Once baked, remove immediately from the muffin pan and set on top of a cooling rack to cool.