Whisk the flour, ginger, cinnamon, baking soda, allspice, salt and pepper in a medium bowl. Beat the butter, molasses and superfine sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce the mixer speed to low; add the flour mixture in 2 batches and beat until just incorporated. Divide the dough in half, wrap in plastic wrap and refrigerate until firm, at least 30 minutes or overnight.
Position racks in the upper and lower thirds of the oven and preheat to 375 degrees F. Working with 1 piece of dough at a time, roll out the dough on a lightly floured surface until about 1/8 inch thick. Cut out shapes using 3-inch cookie cutters and arrange on 2 ungreased baking sheets. Gather the scraps and refrigerate until firm; reroll once to cut out more cookies. Refrigerate the cutouts until firm, about 10 minutes.
Bake the cookies, switching the position of the pans halfway through, until the edges are just browned and set, 12 to 15 minutes. Let cool 2 minutes on the baking sheets, then transfer to racks to cool completely.
Meanwhile, make the icing: Whisk the confectioners' sugar with 1 tablespoon boiling water in a small bowl; add 1 teaspoon water if the icing is too thick. Transfer to a resealable plastic bag; snip off a small corner and pipe the icing onto the cookies. Let set, about 30 minutes.