1.In a stand mixer bowl with a flat beater attached, add granulated sugar and shortening and set to speed 4 until it is light and fluffy. Beat in egg, molasses and vinegar.
2. In a large bowl mix together flour, ground ginger, baking soda, ground cloves and salt. Reduce the speed to stir and mix in the flour mixture, in 3 additions.
3. Divide the dough into 3 disks and wrap with plastic. Refrigerate the dough for at least 3 hours.
4. To prepare the frosting: In a stand mixer bowl with the flat beater attached and set to speed 4, beat butter until it's smooth and creamy. Reduce the speed mix in confectioner's sugar in 2 additions and vanilla. Set frosting aside.
5. Preheat oven to 375 degrees F. Lightly grease baking sheets or line with parchment paper.
6. Flour work surface and rolling pin. Work with one disk at a time. Roll out dough to a 1/8 of an inch thickness.
7. Using a cookie cutter, cut out shapes. Place cookies 2 inches apart on prepared baking sheet. Bake for 5 minutes. Transfer to a cooling rack. Cool cookies completely before frosting.