Whisk the flour, 1/2 cup cocoa powder, the ginger, cinnamon, cloves, allspice, baking soda and salt in a medium bowl. Beat the butter, molasses and brown sugar in a large bowl with a mixer on medium speed until light in color, about 3 minutes. Add the egg and beat until combined. Reduce the mixer speed to low; gradually add the flour mixture, beating until just combined. Divide the dough between 2 sheets of plastic wrap; shape into disks. Wrap and refrigerate until firm, about 2 hours.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Working with 1 disk at a time, roll out the dough on a floured surface until about 1/4 inch thick. Cut out shapes with 3-inch cookie cutters; arrange 1 inch apart on the prepared baking sheets. Reroll the scraps once and cut out more cookies. Refrigerate the cookies until firm, about 15 minutes.
Bake, switching the pans halfway through, until the cookies are set, about 12 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Make the glaze: Whisk the confectioners' sugar, the remaining 1 tablespoon cocoa powder and the orange juice in a bowl. Transfer to a resealable plastic bag and snip a small corner; pipe onto the cookies. Let set, 1 hour.