Step 1: In a large bowl, cream together the softened butter and granulated sugar until the mixture is light, fluffy, and pale in color.
Step 2: Beat in the egg and vanilla extract until fully incorporated. Divide this wet mixture into two equal portions in separate bowls.
Step 3: To the first bowl, sift in 185g of flour and the salt. Mix until a smooth vanilla dough forms. To the second bowl, sift in 165g of flour and the matcha powder. Mix until a vibrant green dough forms.
Step 4: Shape each dough into a rectangular block of equal size (about 1 inch thick). Wrap in plastic wrap and refrigerate for at least 30 minutes until firm.
Step 5: Once firm, cut each block lengthwise into 3 equal strips. You should have 3 vanilla strips and 3 matcha strips.
Step 6: Arrange the strips to create the checkerboard pattern. Lay a vanilla strip next to a matcha strip, then another vanilla strip. Lightly brush the sides with egg white to act as glue. For the second layer, place a matcha strip on top of the vanilla, a vanilla on the matcha, and a matcha on the vanilla.
Step 7: Press the log gently to ensure it sticks together. Wrap tightly in plastic wrap and chill for another 30-60 minutes.
Step 8: Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper.
Step 9: Using a sharp knife, slice the log into cookies about 1/4 inch thick. Place them on the baking sheet 1 inch apart.
Step 10: Bake for 12-15 minutes, or until the edges of the vanilla sections are just beginning to turn a very light golden brown. Cool on the pan for 5 minutes before transferring to a wire rack.