Place the flour, sugar, and salt in a mixer and mix to combine with the whisk attachment.
Add a quarter of the butter and mix until the butter is in tiny pieces.
Add the remaining butter and mix to coat the cubes with flour.
Combine the ice water and lemon juice in a bowl.
Add half the liquid to the flour mixture and mix until just combined.
Keep adding the liquid, 1 Tbsp at a time until the dough starts to clump together.
The dough will be dry at this point.
Turn the dough onto a floured surface and knead.
Press the dough into a 8×4” rectangle.
Wrap it in plastic wrap and refrigerate for at least 30 minutes.
Mix together the sugar and cinnamon.
Lightly flour a large piece of parchment paper.
Place the dough on the parchment and roll it into a 16 by 14 inch rectangle about 1/4 inch thick.
Brush with the beaten egg.
Sprinkle the dough liberally with the cinnamon sugar, gently smoothing the sugar and patting it onto the dough.
With the long side of the dough closest to you, fold 3 inches of the outer edges toward the center, leaving a 2 inch strip in the middle.
Fold the outer edges in again so that both sides of the rectangles are folded in (on top of themselves).
Fold the dough in half, forming a rectangular log.
Wrap in plastic wrap and refrigerate until firm (about 1 hour).
Preheat the oven to 400 degrees.
Slice the chilled roll into 1/2 inch thick slices and position 1 inch apart on a baking sheet.
Bake until the palmiers are golden brown, about 20-24 minutes.
Cool on a cooling rack.