Beat butter, sugar, and salt at medium speed with a heavy-duty electric stand mixer until creamy. Add vanilla; beat until well blended. Gradually add flour, beating at low speed until blended. (Dough will be very soft.)
Divide dough into 2 equal portions; wrap each with plastic wrap, and flatten into a disc. Chill 2 to 24 hours.
Preheat oven to 350°. Unwrap 1 dough disc on a well-floured surface. Cover with a large piece of plastic wrap, and roll to 1/4-inch thickness; cut into desired shapes with 3 1/2-inch cookie cutters. Place 1 inch apart on ungreased baking sheets. Chill 15 minutes. Repeat procedure with remaining dough disc.
Bake at 350° for 14 to 17 minutes or just until edges are lightly browned. Cool on baking sheets 5 minutes. Transfer to wire racks, and let cool completely (about 30 minutes). Decorate cookies with Simple Icing. Let stand 1 hour or until icing is firm.