In a medium bowl, whisk together flour and baking soda.
In a stand mixer fitted with paddle attachment, beat shortening and butter on medium–high speed until smooth, about 30 seconds. Add sugar, molasses, egg, ginger, cinnamon, allspice, cloves, and salt. Beat on medium until combined, scraping bowl as needed, about 1 minute. Beat in vinegar. With mixer on medium, add flour in three increments and beat until combined.
Divide dough in half and transfer to large sheets of plastic. Form each portion into a flat rectangle and wrap tightly in plastic; chill in freezer for at least 1 hour or up to 1 month.
Heat oven to 350ºF (176ºC) with rack set in middle position. Line 2 rimmed baking sheets with parchment paper.
Working with 1 rectangle of dough at a time, cut dough in half, rewrap other half in plastic and place back in freezer.
Lightly dust counter and dough with flour. Roll dough to ¼-inch thickness (flipping dough and adding flour as needed to keep it from sticking to counter). Cut out cookies using a 3- to 4-inch cookie cutter. Peel away scraps and space 6 cookies, about 1 inch apart, on one prepared baking sheet.
Bake until edges of cookies are set and centers are slightly puffed, 8–10 minutes, rotating halfway through.
Cool cookies on sheets 5 minutes; transfer to wire rack to cool completely.
Repeat rolling, cutting and baking with remaining dough and dough scraps. Keeping dough chilled between batches and allowing sheets to cool completely before placing dough cut outs on sheets.