Heat oven to 350° F. In a large bowl, whisk together the flour, ginger, cinnamon, baking soda, cloves, and salt.
Using an electric mixer, beat the butter and granulated sugar until smooth. Add the molasses and egg and beat until fluffy, about 2 minutes.
Reduce the mixer speed to low and gradually add the flour mixture, mixing until just incorporated. Shape into a disk. Wrap in plastic wrap and refrigerate until firm, at least 30 minutes.
On a floured surface, roll the dough to ¼ inch thick. Using cookie cutters, cut into shapes; place on parchment-lined baking sheets.
Bake until firm, 10 to 12 minutes. Cool slightly on baking sheets, then transfer to wire racks to cool completely.
Place confectioners’ sugar in a small bowl; gradually whisk in 1 tablespoon water until a thick icing forms. Transfer to a resealable plastic bag and snip a small hole in one corner. Decorate cookies as desired and let set.
Store in an airtight container at room temperature for up to 1 week.
Total time includes cooling time