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.
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.
Divide the dough into 3 disks and wrap with plastic. Refrigerate the dough for at least 3 hours.
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.
Preheat oven to 375 F.
Flour work surface and rolling pin. Work with one disk at a time. Roll out dough to a 1/8 of an inch thickness.
Using a cookie cutter, cut out your shapes and place them at least 2 inches apart on a baking sheet. Cookies will take about 5 minutes to cook. Transfer to a cooling rack.
Make sure the cookies are completely cooled before frosting.