In a large bowl, cream vegetable shortening and sugar. Beat in eggs, lemon or almond extract, and vanilla extract until mixture is fluffy. Stir in sourdough starter; set aside.
In a medium bowl, stir together flour, baking soda, and salt; stir into sourdough mixture. Refrigerate dough, covered, at least 1 hour or overnight.
Preheat oven to 350 degrees F.
On a lightly floured board, roll dough to 1/4-inch thick; cut into desired shapes with cookie cutters. Place 1 inch apart onto ungreased cookie sheets.
Bake 8 to 10 minutes or until very lightly browned on the bottoms. Remove from oven and cool on wire racks. When cool, frost with Lemon-Milk Frosting and decorate as desired.
In a small bowl, combine powdered sugar, lemon extract, and enough milk to make frosting easy to spread. Tint, if desired, with a few drops of food coloring.
NOTE: Depending on the size of your cookies, you might need to double the frosting recipe.
* If you do not presently have a sourdough starter, either make your own sourdough starter or purchase Packaged Sourdough Starter Mix by mail-order.
Sourdough Sugar Cookie Recipe: https://whatscookingamerica.net/Cookie/SourSugarCookies.htm