Beat butter at medium speed with an electric mixer until creamy. Gradually add sugar, beating well. Add egg and vanilla, beating well.
Combine flour, baking powder, and salt in a separate bowl; add to butter mixture, beating just until blended. Divide dough into 4 portions. Wrap each portion in plastic wrap, and chill at least 1 hour.
Roll each portion to 1/8-inch thickness between 2 sheets of wax paper. Cut with a 5-inch star-shaped cookie cutter. Place 1 inch apart on greased cookie sheets. Bake at 375° for 7 minutes or until edges lightly brown. Cool 3 minutes on cookie sheets; carefully remove to wire racks, and cool completely. Spread or decoratively pipe cookies with Royal Frosting.