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In stand mixer fitted with paddle attachment, cream the shortening and sugars together at medium-high speed until well combined, stopping as necessary to scrape down sides of bowl. With mixer set at medium speed, blend in the lightly beaten egg followed by the molasses and vanilla.
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Mix the baking soda into the hot water in a measuring cup. Add the vinegar and, with mixer set to low speed, transfer the liquid mixture to the batter. Once incorporated, increase mixer speed to medium and beat mixture for just a few seconds until ingredients are fully blended.
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Sift or sieve the flour, cream of tartar, spices, and salt together into a bowl. With mixer set to lowest speed, add the dry ingredients, about ¾ cup at a time, mixing well after each addition and stopping to scrape down sides of bowl, when necessary. Increase mixer speed to medium and mix batter for a few seconds to ensure all ingredients are blended in. Dough will be sticky but do not add any extra flour which would make the cookies, dense, dry, and hard.
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Cover bowl tightly with cling wrap (or transfer to bowl with airtight cover) and refrigerate dough for about 4 hours.
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When ready to bake cookies, position oven rack in center of oven and preheat oven to 350°F. Line cookie sheet(s) with parchment paper. Remove cookie dough from refrigerator and turn out onto a floured work surface. Roll dough to ¼“ thickness. Using a 2” lightly floured cookie cutter, cut out cookies and, using a thin-edged offset spatula, carefully transfer them to the prepared baking sheet(s), spacing cookies about 2½“ apart. Lightly re-flour cutter before cutting out each cookie. Gather dough scraps and re-roll to cut out remaining cookies. Bake for approximately 13-16 minutes, or until cookies appear set and show signs of starting to crack on tops. Turn cookie sheet(s) partway through baking. Adjust baking time according to how your oven heats but do not overbake which will result in hard cookies.
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Remove cookies from oven and allow them to rest on cookie sheet(s) for 5 minutes before transferring them to wire cooling racks to cool completely. Store at room temperature in airtight container to keep the cookies soft and prevent them from drying out. Cookies also freeze well for longer storage.