In a large mixing bowl, beat the butter and salt until creamy, 1 to 2 minutes. Add the egg yolk and vanilla extract and continue to beat to combine well. Sift in the flour and powdered sugar and beat on medium-low until a cohesive dough forms, 1 to 2 minutes. Shape the dough into a ball, flatten it into a thick disk, and divide equally in half. Place half of the dough back in the mixing bowl and sift in the cocoa powder. Mix on medium-low until the dough is a uniform chocolate color.
Shape the plain dough into a thick log and lay it on a large sheet of plastic wrap. Wrap the dough and shape it into a rectangle with square ends, 3 1⁄2 inches long by 1 1⁄2 inches wide; dropping the dough onto a flat surface or using a flat spatula can help shape the sides. Repeat with the chocolate dough, place both dough blocks on a plate, and put the plate into the refrigerator to chill for 1 hour.
Remove the dough from the refrigerator. Unwrap it and use a large sharp knife to cut both blocks in half lengthwise. Lay the chocolate halves on the plain halves, putting the cut sides together and lining them up on all sides. Cut both blocks lengthwise again. Place the cut halves together as before, this time making sure the chocolate and plain are alternating and that the corners line up to create a checkerboard pattern on the ends. Wrap the blocks tightly in plastic and reshape as needed. Place back in the refrigerator for at least 2 hours, or overnight.
Line a large baking sheet with parchment paper. Remove the cookie dough from the refrigerator and unwrap. Using a long, sharp knife, slice the sides of the blocks very thinly to flatten and even them out into concise rectangles. Next, slice the blocks into ¼-inch-thick squares, rotating the blocks after each cut to keep them uniform. Lay the cookie slices on the baking sheet at least 1 inch apart. Place the baking sheet in the refrigerator while you preheat your oven to 350°F.
Slide the baking sheet onto the middle oven rack and bake for 10 to 12 minutes, or until the bottoms are just beginning to turn golden. Remove from the oven and let the cookies sit on the baking sheet for a minute before transferring them to a cooling rack.
