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1
In a medium bowl, whisk the flour with the cocoa powder, ginger, cinnamon, cloves, baking soda, baking powder, and salt. In the bowl of a standing electric mixer fitted with the paddle, beat the softened butter with the shortening at medium speed until the mixture is smooth, about 30 seconds. Add the sugar and beat until fluffy, about 2 minutes.
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2
Add the egg to the cookie batter and beat until incorporated. Beat in the molasses and then the melted chocolate. Add the flour mixture in three batches, beating between additions. Divide the dough into three equal parts. Shape each part into a disk, then wrap each one in plastic wrap and refrigerate the cookie dough until chilled, about 2 hours.
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3
Preheat the oven to 350u00b0F. Line two large baking sheets with parchment paper. On a lightly floured work surface, roll out one disk of dough 1/4-inch thick. Using 4- to 5-inch cookie cutters, cut the dough into shapes and transfer to the prepared baking sheets. Reroll the dough scraps and cut out more cookies. Bake the cookies for about 7 minutes, rotating the pans halfway through the baking time until the tops are dry. Let the cookies cool on the pans for 5 minutes, then transfer to wire racks to cool completely. Repeat the process with the remaining dough.
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4
In a medium bowl, combine the confectioners' sugar with the egg white and lemon juice and whisk until the icing is completely smooth. Scrape the icing into a piping bag fitted with a small tip. Decorate the cookies as desired. Let stand until the icing dries, about 30 minutes. Store tightly covered for 3 days.
