Beat shortening in a mixing bowl with an electric mixer on medium to high speed for 30 seconds. Add sugar, baking powder, ginger, baking soda, cinnamon, and cloves. Beat until combined. Beat in molasses, egg, and vinegar until combined. Beat in as much of the flour as you can with the mixer. Stir in remaining flour with a wooden spoon. Divide dough in half. Cover and chill about 3 hours or until easy to handle.
Roll each portion of dough on a lightly floured surface until 1/8 inch thick. Cut into desired shapes. Place cutouts 1 inch apart on a lightly greased cookie sheet.
Bake in a 375 degree F oven for 5 to 6 minutes or until edges are lightly browned. Cool 1 minute on cookie sheet. Transfer to wire racks to cool. Meanwhile, for the powdered sugar icing stir in powdered sugar and 2 tablespoons milk in a bowl. Add additional milk, 1 teaspoon at a time, until icing reaches drizzling consistency. Color icing with paste food coloring. Spread over cookies. If desired, add a small candy to each cookie to indicate the state capital. Makes about 36.