Line 2 half-sheet pans with parchment paper. Dust the work surface well with flour. Place the dough on the work surface with the open seam of dough facing you. Dust the top of the dough with flour. Using a large, heavy rolling pin, roll out the dough into a 16 by 12-inch rectangle. Don't press the dough too hard; let the weight of the pin do much of the work. If you changed the position of the dough while rolling, it is important to keep track of which side contains the seam.
Turn the dough with the seam facing you. (If you've lost track, look carefully at the sides of the dough, and you should be able to discern it, even though it is faint.) Using a pizza wheel and a yardstick, neatly trim the rough edges. Cut the dough in half lengthwise to make two 16 by 6-inch rectangles. Fold each rectangle into thirds, place on a half-sheet pan, and refrigerate uncovered for about 15 minutes.
Remove the dough strips from the refrigerator. Work with one dough strip at a time. Using a pizza wheel, starting at the top left corner of one strip, cut down to make a half-triangle with a 2-inch base; set aside. Measure 3 1/2-inches from the top left corner of the strip and mark a notch with the wheel at this point. Cut down diagonally from the notch to meet the bottom left edge of the dough strip to make another triangle with a 3 1/2-inch base. Continue cutting, alternating diagonal cuts, to cut out 6 triangles. The last cut will also yield a half-triangle with a 2-inch-wide base. Repeat with the second strip of dough to make 6 more large triangles and 2 half-triangles. You should have a total of 12 large triangles and 4 half-triangles.
Place a single "complete" triangle on the work surface with the base of the triangle facing you. Stretch the bottom slightly so it is about 5-inches wide. Pick up the triangle. With one hand, hold the dough triangle at the bottom and stretch it with your other hand until it is about 7-inches long. Return the triangle to the work surface. Starting at the bottom, roll up the triangle, and finish with the tip underneath the croissant on the pan. Curve the croissant by bringing the 2 ends together and then cross one end over the other, and press together. Repeat rolling the remaining dough triangles, placing them 1 1/2-inches apart on the pan. Overlap 2 of the half-triangles at their long sides, and press the seam together. Roll up as described for the large triangles and add to the pan. Repeat with the remaining half-triangles.
Choose a warm place in the kitchen for proofing. Slip each pan into a tall "kitchen-sized" plastic bag. Place a tall glass of very hot water near the center of each pan. Wave the opening of each bag to trap air and inflate it like a balloon to create "head room," being sure that the plastic does not touch the delicate dough. Twist each bag closed. Let stand until the croissants look puffy but not doubled, 1 1/2 to 2 hours.
Meanwhile, position racks in the center and top third of the oven and preheat to 375 degrees F. Remove the glasses from the bags, then the pans. Lightly brush the croissants with the beaten egg. Bake for 10 minutes. Reduce the heat to 350 degrees F and continue baking until the croissants are crisp and golden brown, about 15 minutes longer. Serve warm or cool to room temperature.
To make the detrempe:
Finely crumble the yeast into the bowl of a heavy-duty stand mixer. Add the sugar and let stand until the yeast gives off some moisture, about 3 minutes. Whisk well to dissolve the yeast. Stir in the milk. (If using dry yeast, sprinkle the yeast over 1/4 cup warm, 105 degrees F to 115 degrees F, milk in a small bowl. Let stand until the yeast softens, about 5 minutes. Whisk well to dissolve. Pour into the mixer bowl, then add the sugar. Add the remaining 1 cup cold milk.)
Mix the bread and pastry flours together. Add 2 cups of the flour mixture and the salt to the bowl. Attach the bowl to the mixer and fit with the paddle attachment. Mix on low speed, adding enough of the remaining flour mixture to make a soft, sticky dough. Do not overmix, as the dough will be worked and absorb more flour during the rolling and folding processes. Transfer the dough to a floured work surface, knead a few times to smooth the surface, and shape into a ball. The ball will hold its shape but spread slightly during standing.
Dust a half-sheet pan with flour. Place the dough on the flour and cut an X about 1-inch deep in the top of the ball to mark it into quadrants. Sprinkle with flour on top and refrigerate.
To make the beurrage:
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. Transfer to a lightly floured work surface and press any remaining lumps of butter out with the heel of your hand, and shape the butter into a 4-inch square. Place the beurrage on the half-sheet pan with the detrempe and refrigerate together 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 dough 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 of the dough with flour. Using the heel of your hand, flatten and stretch each quadrant out about 2 1/2-inches to make a cloverleaf shape with an area in the center that is thicker than the "leaves". Use a tapered rolling pin to roll each "cloverleaf" into a flap about 6-inches long and 5-inches wide, leaving a raised square in the center. Using the side of the rolling pin, press the sides of the raised area to demark 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 don't tear it.) Now stretch and pull the south-facing flap of dough up to cover the top and sides of the butter square. Turn the dough so the open ends of the square face north and south. Repeat folding and stretching the north- and south-facing flaps of dough (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 dough over so the 4 folded flaps face down with the open seam facing you. Dust the top of the dough with flour. Using a large, heavy rolling pin held at a slight angle, lightly pound the top of the dough to widen it slightly and help distribute the butter inside the dough. Roll the dough into a 17 by 9-inch rectangle. Fold the dough into thirds, like a business letter, brushing away 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 of dough facing you. Dust the dough with flour. Roll out the dough into a 17 by 9-inch rectangle. Fold the right side of the dough over 2 inches 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 rolling and folding the dough into a final single turn. With the long seam facing you, cut the dough in half vertically. Wrap each piece of dough tightly in plastic wrap, then wrap again. Freeze for at least 24 hours or up to 4 days.
The night before using the dough, transfer the frozen dough to 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.
Yield: Makes about 2 pounds, 6 ounces dough.
Make the dough at least 2 days before using.
Recipe courtesy Sarabeth Levine, from Sarabeth's Bakery: From My Hands to Yours