Whisk the dry ingredients. In a large mixing bowl, whisk together the flour, baking powder, and salt until combined.
Cream the butter. In a large bowl, use a hand mixer or a stand mixer fitted with a paddle attachment and beat the butter on high until soft and creamy. Then sift in the powdered sugar and granulated sugar and beat on high until creamed about 2 minutes. Then add the egg, egg yolk, and vanilla. Continue beating at high speed until combined. Scrape down the sides of the bowl and beat again to combine for about 2 minutes.
Combine. Add the dry ingredients to the wet ingredients and beat on low to combine. The dough will be thick and sticky.
Divide the dough. Divide the dough in half. Wrap 1/2 half of the dough with plastic wrap into a 6-inch long x 2-inch thick rectangle block.
Make matcha dough. For the other half of the dough, sift in the matcha powder into the mixing bowl. Beat the dough on low speed until combined. Wrap the dough with plastic wrap into a 6-inch long x 2-inch thick rectangle block.
Chill in the refrigerator. Chill both doughs in the refrigerator for 1 hour.
Roll out the dough and cut it into strips. Using two levelers, roll out each dough into rectangles about 5 inches wide x 8 inches long and 1/2 inch thick. Use a ruler to mark 1/2 inch across the dough and then slice into 1/2 inch strips. You should yield about 10 strips for each dough. (2 strips will be leftover)
Make the "glue". In a small bowl whisk together the water and egg white. Set aside.
Assemble. Alternate the dough strips and brush the sides of the dough with glue to hold the dough in place. Continue to top each with a row of alternating flavored dough strips and then again with another row of alternating dough strips remembering to brush the glue on the edges. For example, in the first row, I put 1 vanilla strip, 1 matcha strip, and 1 vanilla strip. Then the second row, I placed 1 matcha strip, 1 vanilla strip, and 1 matcha strip. In the third row, I placed 1 vanilla strip, 1 matcha, and 1 vanilla. As you place the dough slightly press the dough together to get rid of any gaps or air bubbles.
Wrap and chill. Carefully wrap the block of dough and chill for an additional 2 hours.
Prepare the oven and baking sheet. Preheat oven to 350°F (177°C). Line 2 baking sheets with parchment paper or silicone baking mats. Set aside.
Slice and bake. Slice the block about 1/2-inch thick. Arrange the cookies 2 inches apart on the baking sheet. Bake for 13-14 minutes or until very lightly browned on the edges.
Cool. Remove from the oven and allow the cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.