Beat butter and peanut butter at medium speed of an electric mixer until creamy; gradually add sugar, beating well. Add egg and vanilla; beat well.
Combine flour, soda, and salt; add to butter mixture, beating well. Cover and chill dough 1 hour.
Shape dough into 24 balls; place half of dough balls 2" apart on a lightly greased cookie sheet. Dip a fork in water and flatten cookies in a crisscross pattern.
Bake at 350° for 10 minutes. Cool slightly on cookie sheet; transfer to a wire rack. Let cool.
Place remaining dough balls on greased cookie sheet; repeat crisscross procedure with fork. Bake at 350° for 8 minutes. Let cool slightly; turn cookies over on cookie sheet, so crisscross pattern is down. Place a peanut butter cup candy on each cookie.
Bake an additional 2 minutes or until candies soften. Remove from oven. Top each with a baked cookie, pressing to form sandwiches.