1. Combine butter and shortening in a big mixing bowl. Beat at medium speed with an electric mixer until creamy; gradually add sugar, beating well. Add egg, beating well. Stir in peppermint and vanilla extracts. Combine flour, baking powder, and salt; add to butter mixture, beating well. Remove half of dough from bowl. Add food coloring paste to dough in bowl, and mix until color is evenly distributed.
2. Shape plain dough by teaspoonfuls into 4-inch ropes. Do the same thing with red dough. Place 1 red rope and 1 plain rope side by side; gently twist together. Roll twisted ropes into 1 rope. Shape into a cane, and place on an ungreased baking sheet. Do the same thing all over again with remaining ropes. Chill 15 minutes.
3. Preheat oven to 375°.
4. Bake at 375° for 8 minutes or just until lightly brown. Remove baking sheets from oven using oven mitts. Cool slightly. Carefully remove cookies to wire racks, using a spatula. Cool completely. Store cookies in an airtight container up to 1 week.