Clean the mixer bowl and paddle attachment. Add the butter to the bowl and beat with the paddle attachment on medium speed until the butter is almost smooth, about 30 seconds. Add the flour and continue beating until the mixture is smooth, cool, and malleable, about 30 seconds more. This is your beurrage (beurre = butter in French).
Transfer the beurrage to a lightly floured work surface and press any remaining lumps of butter out with the heel of your hand. Shape the beurrage into a 4-inch (10-cm) square, place it on the half-sheet pan with the detrempe, and refrigerate for about 15 minutes. The detrempe and the beurrage should be the same consistency and temperature after this slight chilling.
Flour the work surface again. Place the detrempe on the work surface with the ends of the X at approximately 2, 4, 7, and 10 o'clock positions. You will notice 4 quadrants of dough between the crosses of the X at the north, south, east, and west positions. Dust the top with flour. Using the heel of your hand, flatten and stretch each quadrant out about 2 1/2 inches (7 cm) to make a cloverleaf shape with an area in the center that is thicker than the "leaves."
Use a rolling pin to roll each "cloverleaf" into a flap about 6-inches (15-cm) long and 5-inches (13-cm) wide, leaving a raised square in the center. Using the side of the rolling pin, press the sides of the raised area to mark the square.
Place the butter square in the center of the cloverleaf. Gently stretch and pull the north-facing flap of dough down to cover the top and sides of the butter square, brushing away any excess flour. (This dough is very extendable and will stretch easily but you do need to be a little careful not to tear it. If you do, simply try to pat it back together again.)
Now stretch and pull the south-facing flap up to cover the top and sides of the butter square. Turn the packet so the open ends of the square face north and south. Repeat folding and stretching the north- and south-facing flaps (originally the east and west flaps) to completely cover the butter square, making a butter-filled packet of dough about 6-inches square.
Dust the work surface with flour. Turn the packet over so the 4 folded flaps face down with the open seam facing you. Dust the top with flour. Using a large, heavy rolling pin held at a slight angle, lightly pound the top to widen it slightly and help distribute the butter inside. Roll the dough into a 17-by-9-inch (43-by-23-cm) rectangle. Fold the dough into thirds, like a business letter, brushing away any excess flour. This is called a single turn. Roll the rectangle lightly to barely compress the layers. Transfer to a half-sheet pan and refrigerate for about 20 minutes.
Lightly flour the work surface. Place the dough on the work surface with the long open seam facing you. Dust the top with flour. Roll out the dough into a 17-by-9-inch rectangle. Fold the right side of the dough over 2 inches (5 cm) to the left. Fold the left side of the dough over to meet the right side. Fold the dough in half vertically from left to right. This is a double turn (also known as a book turn). Roll the rectangle lightly to barely compress the layers. Return to the half-sheet pan and refrigerate for another 20 minutes.
Repeat the last rolling and folding into a final single turn. With the long seam facing you, cut the dough in half vertically. Wrap each piece tightly in plastic wrap, then wrap again. Freeze for at least 24 hours or up to 4 days. The night before using the dough, place it in the refrigerator and let thaw overnight, about 8 hours. Once defrosted the dough will begin to rise, so make sure to roll it out immediately.