Beat vegetable shortening, white sugar, 1/4 cup milk, eggs, and 1 teaspoon vanilla extract together in a bowl. Whisk flour, salt, and baking soda in a separate bowl. Slowly beat flour mixture into shortening mixture to make a smooth dough.
Roll dough out 1/4-inch thick on a floured work surface and cut into shapes. Arrange cookies on baking sheets.
Bake in the preheated oven until cookies are firm, about 10 minutes. Let cool.
Beat butter, confectioners' sugar, 1 1/2 teaspoon vanilla extract, and 2 tablespoons milk in a bowl with an electric mixer on high speed until frosting is spreadable and slightly fluffy. Frost cooled cookies.
