In a medium mixing bowl, stir together flour, cocoa powder, baking soda, and salt; set aside. In a large mixing bowl, beat butter with an electric mixer on medium to high speed 30 seconds.
Add the brown sugar; beat until combined, scraping bowl occasionally. Add egg, molasses, and vanilla and beat until combined. Beat in as much of the flour mixture as you can with the mixer. Stir in any remaining flour mixture with a wooden spoon. Divide dough in half. Cover; chill at least 24 hours or until easy to handle.
Preheat oven to 375F. On a lightly floured surface, roll each dough portion to about 1/4-inch thickness. With spider- and web-shaped cutters, cut into shapes. Place shapes 1 inch apart on ungreased cookie sheets. (Dip cutter in flour between cutting out each cookie. This helps prevent the dough from sticking to the cutter.)
Bake in preheated oven for 7 to 9 minutes for spider cookies (9 to 11 minutes for web-shaped cookies) or until edges are firm and bottoms are light brown. Let cool on cookie sheets for 1 minute. Carefully remove cookies; cool thoroughly on wire racks. Decorate with tinted Powdered Sugar Icing.
Powdered Sugar Icing
In a medium mixing bowl, stir together powdered sugar, melted butter, and vanilla. Add milk, 1 teaspoon at a time, until icing reaches spreading consistency. Divide icing into portions and tint as desired.