Preheat the oven to 350 degrees and line a baking sheet with parchment paper.
Stir together the flour, cocoa powder, baking soda, and salt in a mixing bowl.
Cream the butter and granulated sugar until light, then beat in the egg and vanilla extract.
Add the dry ingredients to the wet mixture and mix until a dough forms.
Roll the dough to 1/4 inch thick, cut into shapes, and place on the baking sheet.
Bake for 8 to 10 minutes until the edges are set, then cool completely on a wire rack.
Whisk the powdered sugar with the milk to make a smooth white icing.
Divide a small portion of the icing and tint it black with food coloring.
Pipe white circles on each cookie, then add thin black lines and drag a toothpick through them to form spiderwebs.