Cream butter; gradually add sugar, beating well. Add egg; beat well. Sift together flour, baking powder, and salt; add to creamed mixture, beating well. Stir in sesame seeds and vanilla; mix well.
Drop dough by teaspoonfuls onto waxed paper-lined baking sheets. Bake at 325° for 10 minutes or until edges are lightly browned. Cool 5 minutes. Carefully remove to wire racks to cool completely.