1. Beat butter at medium speed with a heavy-duty electric stand mixer until creamy; gradually add sugar, beating until light and fluffy. Add egg and vanilla, beating until blended, scraping bowl as needed.
2. Combine flour, baking soda, and salt; gradually add flour mixture to butter mixture, beating at low speed until blended.
3. Divide dough into 2 equal portions. Roll 1 portion of dough into a 12- x 8-inch rectangle on a piece of lightly floured plastic wrap.
4. Knead food coloring paste into remaining portion of dough while wearing rubber gloves. Roll tinted dough into a rectangle as directed in Step 3. Invert untinted dough onto tinted dough; peel off plastic wrap. Cut dough in half lengthwise, forming 2 (12- x 4-inch) rectangles. Roll up each rectangle, jelly-roll fashion, starting at 1 long side, using bottom piece of plastic wrap as a guide. Wrap in plastic wrap, and freeze 4 hours or up to 1 month.
5. Preheat oven to 350°. Cut ends off each dough log, and discard. Cut dough into 1/4-inch-thick pieces, and place on parchment paper-lined baking sheets.
6. Bake at 350° for 6 to 7 minutes or until puffed and set; cool cookies on baking sheets 5 minutes. Remove to wire racks, and cool completely (about 30 minutes).
7. Place Peppermint Frosting in a heavy-duty zip-top plastic freezer bag. Snip 1 corner of bag to make a small hole. Pipe about 2 tsp. frosting onto half of cookies; top with remaining cookies, gently pressing to form a sandwich.