Preheat oven to 350°. Beat butter, vanilla, and sugar at medium speed with an electric mixer until fluffy. Add egg, beating until blended.
Combine flour and salt; gradually add to butter mixture, beating just until blended.
Divide dough into 2 equal portions; flatten each portion into a disk. Cover and chill 10 minutes.
Place 1 portion of dough on a lightly floured surface, and roll to 1/8-inch thickness. Cut with a 3- or 4 1/2-inch butterfly-shaped cutter. Place cookies 2 inches apart on lightly greased baking sheets. Repeat procedure with remaining dough portion.
Bake, in batches, at 350° for 10 to 12 minutes (for 3-inch cookies) and 15 to 17 minutes (for 4 1/2-inch cookies) or until edges are lightly browned. Let cool on baking sheets 5 minutes; remove to wire racks, and let cool 30 minutes or until completely cool. Decorate baked cookies as desired.