Combine the flour through the salt in the bowl of a stand mixer fitted with the dough hook. Mix on low speed for 3 minutes, scraping the sides of the mixing bowl once if necessary. Mix on medium speed for 3 minutes. Transfer the dough to a lightly floured 10-inch pie pan or a dinner plate. Lightly flour the top of the dough and wrap well with plastic so it doesn’t dry out. Refrigerate overnight.
The next day, cut the cold butter lengthwise into 1/2-inch-thick slabs. Arrange the pieces on a piece of parchment or waxed paper to form a 5- to 6-inch square, cutting the butter crosswise as necessary to fit. Top with another piece of parchment or waxed paper. With a rolling pin, pound the butter with light, even strokes. As the pieces begin to adhere, use more force. Pound the butter until it’s about 7-1/2 inches square and then trim the edges of the butter. Put the trimmings on top of the square and pound them in lightly with the rolling pin. Refrigerate while you roll out the dough.
Unwrap and lay the dough on a lightly floured work surface. Roll into a 10-1/2-inch square. Brush excess flour off the dough. Remove the butter from the refrigerator—it should be pliable but cold. If not, refrigerate a bit longer. Unwrap and place the butter on the dough so that the points of the butter square are centered along the sides of the dough. Fold one flap of dough over the butter toward you, stretching it slightly so that the point just reaches the center of the butter. Repeat with the other flaps . Then press the edges together to completely seal the butter inside the dough. (A complete seal ensures butter won’t escape.)
Lightly flour the top and bottom of the dough. With the rolling pin, firmly press the dough to elongate it slightly and then begin rolling instead of pressing, focusing on lengthening rather than widening the dough and keeping the edges straight.
Roll the dough until it’s 8 by 24 inches. If the ends lose their square shape, gently reshape the corners with your hands. Brush any flour off the dough. Pick up one short end of the dough and fold it back over the dough, leaving one-third of the other end of dough exposed. Brush the flour off and then fold the exposed dough over the folded side. Put the dough on a baking sheet, cover with plastic wrap, and freeze for 20 minutes to relax and chill the dough.
Repeat the rolling and folding, this time rolling in the direction of the two open ends until the dough is about 8 by 24 inches. Fold the dough in thirds again, as shown in the photo above, brushing off excess flour and turning under any rounded edges or short ends with exposed or smeared layers. Cover and freeze for another 20 minutes.
Give the dough a third rolling and folding. Put the dough on the baking sheet and cover with plastic wrap, tucking the plastic under all four sides. Refrigerate overnight.
The next day, unwrap and lightly flour the top and bottom of the dough. With the rolling pin, “wake the dough up” by pressing firmly along its length—you don’t want to widen the dough but simply begin to lengthen it with these first strokes. Roll the dough into a long and narrow strip, 8 inches by about 44 inches. If the dough sticks as you roll, sprinkle with flour. Once the dough is about half to two-thirds of its final length, it may start to resist rolling and even shrink back. If this happens, fold the dough in thirds, cover, and refrigerate for about 10 minutes; then unfold the dough and finish rolling. Lift the dough an inch or so off the table at its midpoint and allow it to shrink from both sides—this helps prevent the dough from shrinking when it’s cut. Check that there’s enough excess dough on either end to allow you to trim the ends so they’re straight and the strip of dough is 40 inches long. Trim the dough.
Lay a yardstick or tape measure lengthwise along the top of the dough. With a knife, mark the top of the dough at 5-inch intervals along the length (there will be 7 marks in all). Position the yardstick along the bottom of the dough. Make a mark 2-1/2 inches in from the end of the dough. Make marks at 5-inch intervals from this point all along the bottom of the dough. You’ll have 8 marks that fall halfway between the marks at the top.
Make diagonal cuts by positioning the yardstick at the top corner and the first bottom mark. With a knife or pizza wheel, cut the dough along this line. Move the yardstick to the next set of marks and cut. Repeat until you have cut the dough diagonally at the same angle along its entire length—you’ll have made 8 cuts. Now change the angle of the yardstick to connect the other top corner and bottom mark and cut the dough along this line to make triangles. Repeat along the entire length of dough. You’ll end up with 15 triangles and a small scrap of dough at each end.
Using a paring knife or a bench knife, make a 1/2- to 3/4-inch-long notch in the center of the short side of each triangle. The notch helps the rolled croissant curl into a crescent. Hold a dough triangle so that the short notched side is on top and gently elongate to about 10 inches without squeezing or compressing the dough—this step results in more layers and loft.
Lay the croissant on the work surface with the notched side closest to you. With one hand on each side of the notch, begin to roll the dough away from you, towards the pointed end.
Flare your hands outward as you roll so that the “legs” become longer. Press down on the dough with enough force to make the layers stick together, but avoid excess compression, which could smear the layers. Roll the dough all the way down its length until the pointed end of the triangle is directly underneath the croissant. Now bend the two legs towards you to form a tight crescent shape and gently press the tips of the legs together (they’ll come apart while proofing but keep their crescent shape).
Shape the remaining croissants in the same manner, arranging them on two large parchment-lined rimmed baking sheets (8 on one pan and 7 on the other). Keep as much space as possible between them, as they will rise during the final proofing and again when baked.
Make the egg wash by whisking the egg with 1 tsp. water in a small bowl until very smooth. Lightly brush it on each croissant.
Refrigerate the remaining egg wash (you’ll need it again). Put the croissants in a draft-free spot at 75° to 80°F. Wherever you proof them, be sure the temperature is not so warm that the butter melts out of the dough. They will take 1-1/2 to 2 hours to fully proof. You’ll know they’re ready if you can see the layers of dough when the croissants are viewed from the side, and if you shake the sheets, the croissants will wiggle. Finally, the croissants will be distinctly larger (though not doubled) than they were when first shaped.
Shortly before the croissants are fully proofed, position racks in the top and lower thirds of the oven and heat it to 400°F convection, or 425°F conventional. Brush the croissants with egg wash a second time. Put the sheets in the oven. After 10 minutes, rotate the sheets and swap their positions. Continue baking until the bottoms are an even brown, the tops richly browned, and the edges show signs of coloring, another 8 to 10 minutes.

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