Mix butter, sugar and vanilla with electric mixer fitted with paddle on low speed until creamy and smooth. Add flour and salt; mix on low until well blended. Scrape bowl; mix on low 20 seconds. Add OREO Pieces; mix on low 20 seconds. Scrape bowl; mix on low 20 seconds.
Roll out dough on lightly floured surface to 1/4-inch thickness, keeping one edge at exactly 12 inches. Cut out strips 6 inches long and 1/2-inch wide. (Dough can be re-rolled once. Do not overwork). Arrange strips 1/2-inch apart on parchment paper-lined sheet pans.
Bake in 350°F standard oven 20 to 23 min. or until light golden brown on edges. Cool completely in pan on wire rack. Plate 2 "fries" per serving.