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 1/3-inch thickness. Cut with a 21/2 x 5 1/2-inch pennant-shaped cutter. Place 6 cookies on each lightly greased baking sheet. Insert 1 lollipop stick into bottom of each cookie. Repeat procedure with remaining dough.
Bake, in batches, at 350° for 12 to 14 minutes 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.