Make the cookies
In a medium bowl, whisk the flour, ginger, baking soda, cinnamon, nutmeg, cloves, and salt. In a stand mixer fitted with the paddle attachment, beat the butter and brown sugar on medium speed until light and fluffy, about 2 minutes. Add the egg, mixing until combined, then add the molasses and orange zest and beat on medium speed until smooth, stopping to scrape the bowl as necessary, about 2 minutes. On low speed, slowly add the dry ingredients and mix just until the flour is incorporated, scraping the bowl as necessary, about 2 minutes. The dough will be very stiff and sticky.
Scrape the dough onto a lightly floured baking sheet and pat it flat to a 1-inch thickness. Cover with plastic wrap and freeze for at least 1 hour and up to 4 weeks.
Position racks in the upper and lower thirds of the oven and heat the oven to 350°F. Line 2 baking sheets with parchment.
Cut the chilled dough into quarters On a well-floured work surface, roll out one piece of dough to 1/8 inch thick (keep the remaining dough in the freezer). If the dough sticks to your work surface, slide an offset spatula under the dough to release it and dust the work surface with more flour.
With a 4-1/4-inch cookie cutter, cut out the cookies and use an offset spatula to transfer them to the baking sheets, spacing them at least 1/2 inch apart. Brush off any excess flour.
Gather and press the dough scraps together, then roll to 1/8 inch thick, and cut out more shapes. Add any remaining scraps to the next quarter of chilled dough. Continue rolling and cutting until the baking sheets are full.
Bake the cookies, rotating and swapping the sheets’ positions halfway through baking, until firm to the touch and the edges are slightly golden, about 10 minutes. Slide the cookies on their parchment onto racks and let cool. Roll out and bake the remaining dough in batches on fresh sheets of parchment.
Make the royal icing
In a stand mixer fitted with the whisk attachment (or in a medium bowl with a hand mixer) mix the confectioners’ sugar, powered egg whites, and cream of tartar on low speed. Increase the speed to medium low and add 3 to 5 Tbs. of water, 1 Tbs. at a time, mixing well and scraping the bowl as necessary, until the mixture is evenly moistened and the whisk leaves small trails that disappear quickly, 2 to 3 minutes. Continue to beat on medium low until the icing is thick, glossy, smooth, and free of lumps, about 5 minutes more.
To test for proper consistency, put a dot of icing on the counter with a toothpick. If, when you pull up the toothpick, the icing forms a peak that slowly melts back into the dot, the icing is ready. If the peak stays up, it’s too stiff: Add 1 tsp. of water to the icing, beat to combine, and test again. Repeat if necessary.
Decorate the cookies
Scrape the icing into a pastry bag fitted with a small tip. (You can also use a plastic squeeze bottle or a plastic sandwich bag with a corner trimmed off.) Decorate the cookies as you like and let sit until the icing is completely set, about 40 minutes.
Store baked and decorated cookies in an airtight container at room temperature for up to 5 days.