Position a rack in the center of the oven and preheat to 375 degrees F. Grease the bottom and sides of a 10-inch cast-iron skillet with 1 tablespoon of the butter. Whisk the flour, ginger, cinnamon, baking soda, cloves and salt together in a medium bowl; set aside.
Beat the remaining butter and sugar together in a large bowl with an electric mixer on medium-high speed until well combined, about 3 minutes. Add the molasses and egg and beat until light in color, about 3 minutes. Add the flour mixture and mix on low speed until well combined.
Use a rubber spatula to scrape the dough into the prepared skillet, gently spreading it to the edges and smoothing the top, then sprinkle with some granulated sugar. Bake until the sides are lightly browned but the center is still quite soft, about 25 minutes. Cool on a wire rack for 15 minutes.
For the icing: Whisk the confectioners' sugar and 1 tablespoon of the milk together until it is thick and stiff enough to pipe or spread. If the icing is too stiff and dry, slowly drizzle in up to 1 more tablespoon milk to reach the proper consistency. Drizzle, spread or pipe the icing onto the cookie, cut into wedges and serve.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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