Cream butter in a medium mixing bowl; gradually add sugar, beating until light and fluffy. Stir in vanilla. Gradually add flour and pecans, stirring just until blended.
Shape dough into a roll, 2 inches in diameter; wrap each roll in waxed paper, and chill several hours until firm.
Cut into 1/4-inch slices. Slice circles in half; place halves 1 inch apart on ungreased cookie sheets. Bake at 375° for 5 minutes. Remove from cookie sheets, and cool on wire racks.