Flour mixture: In a medium bowl, stir together the flour and salt. Set aside.
Butter mixture: In a large mixing bowl, use an electric hand mixer to beat the softened butter until very creamy. Add the granulated sugar and beat until creamy and fluffy. Add the egg yolk and vanilla extract and beat again on low speed until combined.
Combine flour and butter mixtures: Add the flour mixture to the butter mixture and beat on low speed until just barely combined with some flour patches still remaining.
Divide dough: Divide the dough in half and transfer one half to a separate mixing bowl (use the bowl you used for the flour mixture). Finish mixing the dough in one of the bowls just until there are no more flour patches. This will be your plain/vanilla dough. Set aside.
Matcha dough: To the other dough, sift in matcha powder. Use the hand mixer to mix on low speed until the matcha is evenly distributed throughout the dough.
Shape dough: On your kitchen counter, lay down a sheet of parchment paper. Turn out the matcha dough onto the parchment paper. Use your hands to press it into a rectangular shape. Use a bench scraper or something with a straight edge to square off the sides of the rectangle as best you can.
Roll out dough: Lay another sheet of parchment paper on top. Use a rolling pin to roll the dough out to about ¼" thick, keeping it in a rectangle shape as best you can. Occasionally, use a bench scraper to straighten the sides.
Chill dough: On a separate sheet of parchment paper, repeat the previous two steps with the plain dough, making both sheets of dough roughly the same size and shape. Transfer both sheets of dough to the fridge to chill for at least 1 hour or until firm.
Stack dough: Once chilled, peel one sheet of dough off the parchment paper and lay it directly on top of the other sheet of dough.
Stack dough again: Use a knife to cut the stack of dough in half widthwise. Use a pastry brush to brush the top of one half with a thin layer of egg white. Pick up the other half of dough and lay it on top of the egg white layer so that the colours of dough are alternating. Pop this back in the fridge for 15 minutes to firm up again.
Cut strips: Use a sharp knife to cut strips of dough that are about ¼" thick. Each long strip should have 4 stripes of alternating colours and you should end up with 12 strips.
Create checkerboard pattern: Place one strip flat on your surface. Brush the top with a thin layer of egg white. Take another strip and flip it over so that the colours are the opposite of the ones below it. Place the strip down on top of the first strip, lining it up down the length of the strip, and gently press it down to seal together. Repeat until you have a stack of 4 strips. Repeat with the remaining strips. You should end up with 3 "blocks" with 4 strips each. Chill the blocks in the fridge for another 15 minutes to firm up again.
Prep: Preheat the oven to 350°F. Line a baking sheet with parchment paper. Set aside.
Slice cookies: Use a sharp knife to slice each block into ¼" thick cookies. Place the cookies on your lined baking sheet, leaving about ½" in between each cookie. They won't spread much so they can be fairly close together.
Bake: Bake cookies for 12-15 minutes or until the bottom edges of the cookies start to lightly brown.
Cool: Let the cookies cool on the baking sheet for about 5 minutes before transferring them to a wire rack to cool completely.