Preheat oven to 350º.
Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, cocoa, baking soda, and salt, stirring well with a whisk. Place shortening in a large mixing bowl; beat with a heavy-duty mixer at medium speed until light and fluffy. Gradually add the granulated sugar, 1 tablespoon at a time, beating until well blended. Add vanilla and egg white; beat well. Add flour mixture; beat until well blended.
Turn dough out onto wax paper; shape into a 6-inch log. Wrap log in wax paper. Freeze 2 hours or until very firm.
Cut log into 24 (1/4-inch) slices, and place slices 1 inch apart on baking sheets coated with cooking spray. Bake at 350º for 10 minutes or until set. Remove from pans; cool completely on wire racks.
Combine powdered sugar and milk in a medium bowl; stir well with a whisk until smooth. Spoon into a small zip-top plastic bag; seal. Snip a tiny hole in 1 corner of bag. Working with 1 cookie at a time, pipe 3 concentric circles onto each cookie. Starting at center circle, pull a wooden pick through other circles at regular intervals to create a "web."
For the glaze, snip a very small hole in the corner of the plastic bag. If it's too small, you can always make it bigger.