In a large bowl, stir together 1 cup butter and brown sugar until smooth and creamy. Stir in 1 teaspoon vanilla. In a separate bowl, combine the flour, salt, and baking soda; stir into creamed mixture. Mix in milk, stirring until a soft dough forms. Cover, and chill for 1 hour.
Preheat oven to 365 degrees F (185 degrees C). On a floured surface, roll out dough to 1/4 inch thick. Cut into desired shapes using cookie cutters. Place cookies 2 inches apart on ungreased cookie sheets.
Bake in a preheated oven for 6 minutes. Cool on wire racks.
Meanwhile, in a large bowl, stir together 1 cup butter and 1 teaspoon vanilla until it becomes smooth and creamy. Gradually stir in sugar, one cup at a time, beating well after each addition. Beat in a splash of milk, and continue mixing until light and fluffy. Keep icing covered until ready to decorate. Spread icing on top of cookies.