Mix sugar, shortening, buttermilk, and egg together in the bowl of an electric stand mixer (such as KitchenAid®) until creamy.
Combine flour, baking powder, baking soda, vanilla extract, and salt in a bowl. Beat into sugar mixture, 1/2 cup at a time, mixing well after each addition. Leave dough to set in a refrigerator for at least 2 hours to overnight.
Preheat the oven to 325 degrees F (165 degrees C).
Roll dough out on a lightly floured work surface using a lightly floured rolling pin to desired thickness. Cut into shapes using cookie cutters and place onto ungreased cookie sheets.
Bake in the preheated oven until bottoms begin to lightly brown, 8 to 12 minutes. Let cookies cool completely on wax paper or a wire rack, 15 to 30 minutes.
Combine confectioners' sugar and milk in a bowl. Mix into a paste, adding more sugar or milk as needed. Spread frosting thinly onto cooled cookies and cover with sprinkles.