For the Dough:
1 cup milk, heated to 115°
1 tablespoon active dry yeast
3 tablespoons sugar
2 teaspoons kosher salt
3 cups all-purpose flour, plus more for dusting
For the Butter Insert:
¼ cup sugar
2 tablespoons all-purpose flour
1 teaspoon kosher salt
2 sticks (8 ounces) unsalted butter, chilled, plus more for greasing
1 cup sugar, divided
1. Make the dough: In a stand mixer fitted with the dough hook attachment, combine the milk and yeast, and let sit until foamy, 8 to 10 minutes. With the motor running, add the sugar and salt, followed by the flour. Run the mixer until a dough comes together, 3 to 4 minutes. Transfer to a lightly floured bowl and cover. Let sit in a warm place until it has doubled in size, 1 hour.
2. Meanwhile, make the butter insert: Line a baking sheet with parchment paper. Sprinkle the sugar, flour and salt onto a clean work surface and toss the butter in it. Using a rolling pin, beat the butter until pliable, flipping as needed. Roll the butter into a 9-by-12-inch rectangle and transfer it to the prepared baking sheet. Chill for 45 minutes.
3. Transfer the dough to a lightly floured surface. Roll into an 18-by-14-inch rectangle that's ¼ inch thick. Place the chilled butter insert on one half of the dough, then fold the dough over to enclose it, pressing the edges to seal.
4. Assemble the pastries: Grease a muffin tin with butter and sprinkle with 2 tablespoons of sugar, then place the tin on a baking sheet. Working quickly, sprinkle a clean work surface with a heavy pinch of sugar. Place the dough on top and sprinkle with another heavy pinch of sugar. With one of the short ends of the dough facing you, fold the dough like a letter, bringing the bottom third of the dough up over the middle, then the top third of the dough on top of that. Roll the dough into a 12-by-16-inch rectangle, using sugar as you would typically use flour to prevent sticking. Repeat this process 3 more times for a total of 4 folds.
5. After the dough is rolled out for a final time, cut the rectangle into twelve 4-inch squares. With each square, bring all the corners into the center and fit into one of the molds of the muffin tin. Repeat with the remaining squares of dough. Sprinkle the tops with the remaining sugar. Cover in plastic wrap and let sit in a warm place until the dough has expanded to fill each mold, 30 minutes.
6. Preheat the oven to 375°. Bake until golden brown, 25 to 30 minutes. Immediately remove each kouign-amann from the tin and transfer to a wire rack to cool slightly, then serve.