Line a 9x9 inch square baking pan with wax paper. To keep the paper from sliding around, it helps to brush the pan with a little butter first.
Take the disk of vanilla dough, and shape it into a square. Then roll it out into a thick square measuring 8x8 inches. Set the dough into the pan, and finish pressing it out to fill the pan all the way to the edges, trying to keep it as even as possible.
Lift the wax paper with the dough out of the pan, and set it on a cutting board.
Now line the pan with another piece of wax paper, and repeat the above steps with the chocolate dough.
Brush both squares of dough free of excess flour. Dip a pastry brush in water, and lightly brush it over the top of the vanilla dough so that the dough is slightly tacky (but not drenched in water). Then lay the chocolate dough over the top of the vanilla, pressing it down gently to stick the squares of dough together.
Cover the dough with plastic wrap, and freeze for 20 minutes.
Measure the diameter of the chilled dough, then use a straight edge to cut it into 4 equal rectangles.
Again, brush all the dough free of excess flour. Dip a pastry brush in water, and lightly brush it over the top of one section of dough, then stack another section of dough on top. Repeat until you've stacked all four sections of dough and you have a long cube of dough with alternating layers of chocolate and vanilla dough. Gently press down on the top to ensure all the layers are sticking together.
Wrap the dough up in the wax paper, and refrigerate for 40 minutes, then transfer it to the freezer for 20 more minutes before slicing the cookies.
If you want to take this option a step further and make checkerboard cookies, use a large sharp chef's knife to trim the sides of the dough so it's perfectly square and even. Then cut it lengthwise into 8 long slabs. Stack the slabs so that the pattern of the chocolate and vanilla alternates (starting with vanilla on one side for the first slab, then flipping the next slab over so that you start with chocolate, etc.). This way, when you slice the cookies, you'll have a beautiful checkerboard pattern. Use a little water in between each slab to ensure the dough is sticking, press all the layers together, and chill again before slicing into cookies.
After the dough has chilled, you can slice it into cookies. Use a large, sharp chef's knife to slice the cookies 1/4 inch thick. If needed, trim any uneven edges.
Place the cut cookies two inches apart on two baking sheets lined with parchment paper. Refrigerate for 30 minutes before baking.