Mix flour, baking soda and salt in medium bowl. Set aside. Beat sugar and butter in large bowl with electric mixer on medium speed until light and fluffy. Add egg; mix well. Gradually stir in flour mixture on low speed until well mixed. Divide dough in half. Add vanilla to one half; mix well. Add peppermint extract and food color to second half; mix until dough is evenly tinted.
Roll out vanilla dough between wax paper to 15x9-inch rectangle. Repeat with peppermint dough. Peel off the top wax paper from each dough rectangle. With the bottom wax paper still in place, carefully invert vanilla dough onto peppermint dough. Gently press layers together. Peel off the top wax paper. Using the bottom wax paper as a guide, roll dough tightly from the long side into a log. Wrap in plastic wrap. Refrigerate 2 hours or overnight until firm.
Preheat oven to 375°F. Slice log into 1/4-inch thick slices. Place on greased baking sheets.
Bake 8 to 10 minutes or until cookies are set. Cool on baking sheets 1 minute. Remove to wire racks; cool completely.