For the cookies: In the bowl of an electric mixer, cream the shortening and granulated sugar together until light and fluffy. Add the molasses, vinegar and egg and beat on high speed to blend thoroughly.
Sift together the ginger, baking soda, cinnamon, cloves, salt and 4 cups of the flour in a separate bowl. Add the dry ingredients to the creamed shortening and sugar and mix to make a firm, manageable dough, adding more flour if needed. Wrap the dough in plastic and refrigerate until firm, about 3 hours.
Preheat the oven to 375 degrees F. Lightly spray 2 baking sheets with cooking spray.
Spray a rolling pin with cooking spray and lightly flour your work surface. Roll the dough to 1/8-inch thickness and cut it into desired shapes with a knife or cookie cutters. (You can reroll the scraps to make a few more cookies.) Place the cookies 1 inch apart on the prepared baking sheets and bake for 5 to 6 minutes. Cool the cookies for 2 minutes on the baking sheets before removing to a wire rack to cool completely. Repeat with the remaining dough.
For the royal icing: Beat the egg whites and 1/3 cup of the confectioners' sugar with a wire whisk or the whisk attachment of your mixer. Add another 1/3 cup sugar and the cream of tartar and beat 10 minutes longer. Add the remaining sugar and beat until the mixture is smooth and thick. The icing can be tinted as desired with food coloring.
Ice the cooled cookies with the royal icing as desired.
Keep the bowl of icing and the decorator tips covered with a moist cloth to prevent the icing from hardening prematurely.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.