Combine butter, sugar and vanilla essence in a mixing bowl and beat till the mixture becomes light and fluffy.
Add the all purpose flour and mix well. Add milk and knead to make a soft dough.
Divide the dough into 2 equal portions. Add cocoa powder in one part and knead well.
Roll both parts into 1.5 inch thick square-rectangular sticks and wrap them in a plastic wrap.
Cool the sticks in the freezer for about 30 minutes.
Slice each dough lengthwise into 3 pieces of around 1/2 inch thickness.
Again cut each slab lengthwise into 3 pieces of equal width.
Rearrange by swapping the middle dough again. Make two large slabs again by placing the smaller slabs on each other, making sure that alternate colors are stacked.
Wrap each dough in plastic wrap and refrigerate again for 30 minutes.
Slice the slabs crosswise into 1/4 inch thick pieces and place them on a baking tray.
Bake the cookies in a preheated oven at 180 C or 350 F for around 15 minutes.
Allow the cookies to cool on a wire rack. Checkerboard Cookies are ready.