Cream butter and peanut butter in a large mixing bowl. Gradually add sugar, beating well. Beat in egg and vanilla.
Combine flour, baking powder, and salt; add to creamed mixture, stirring well.
Divide dough in half; shape each half into a roll, 2 inches in diameter. Wrap in waxed paper, and chill 3 hours or until firm.
Cut into 1/4-inch slices; place 2 inches apart on ungreased cookie sheets. Bake at 350° for 10 minutes or until lightly browned. Remove from cookie sheets, and cool on wire racks.