Have a yardstick and a pizza wheel at your side to help keep the cuts straight. Always cut from top to bottom. You will have better control over the wheel than if you cut from the bottom up. At the bakery, we combine the half-triangle scraps of dough with trimmings from the Danish dough to make Pains de Matin, sweet breakfast rolls. However, it is unlikely that a home baker will collect enough scraps to make baking these rolls practical. I don't want to waste the croissant trimmings, so I give instructions for piecing the half-triangles together to make full-size croissants. They will not be perfect, but it is possible that no one will notice.
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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.