Shape the dough into a disc and wrap it with cling film. Refrigerate it while you make the chocolate dough.
Make the chocolate dough by repeating the steps of the vanilla dough and adding the cocoa powder with the flour.
1¼ cups unbleached all-purpose flour, 2 tablespoons Dutch-processed cocoa powder, ⅓ cup granulated cane sugar, ¼ cup confectioner’s sugar, ⅛ teaspoon salt, ¼ cup vegan butter or margarine, 1 teaspoon vanilla extract, 4 tablespoons plant milk
Shape the dough into a disc and wrap it with cling film. Place it in the refrigerator and remove the vanilla dough.
Roll out the vanilla dough between two sheets of parchment paper, lightly dusted with flour. Roll the dough to about 8 by 12 inches and ¼-inch thick.
Remove the chocolate dough from the fridge and repeat the rolling step with another sheet of parchment paper.
Remove the parchment paper and carefully place the chocolate dough on top of the vanilla dough. Starting from the shorter side, tightly roll the dough together like a jellyroll. Wrap it tightly in plastic wrap and chill for 2 hours or until firm (overnight is recommended).
Adjust the oven rack to the middle position and preheat to 350°F/180°C. Line cookie sheets with parchment paper.
Slice the chilled dough into ¼-inch thick slices and place them on the cookie sheets spaced 1 inch apart.
Bake the cookies in batches for 8 to 10 minutes or until they are just set and crips on the edges. Cool for 5 minutes on the tray before transferring them to a cooling rack.