In a large bowl beat butter at medium speed until creamy. Add sugar gradually beating until light and fluffy. Scrap down the sides of the bowl and beat in the egg and vanilla.
In a separate bowl whisk together the flour, baking soda, and salt. Add the flour mixture gradually to the butter mixture beating at low speed until well blended.
Divide dough into 2 equal portions. Roll 1 portion of dough into a 12- x 8-inch rectangle on a piece of well-floured parchment paper.
Knead food coloring into the remaining portion of dough while wearing rubber gloves to get a reddish pink color. Roll tinted dough into a 12 x 8 rectangle on a separate piece of well-floured parchment paper. Invert untinted dough onto tinted dough; peel off the parchment.
Cut dough in half lengthwise to form 2 (12- x 4-inch) rectangles. Roll each one up starting at 1 long side, using bottom piece of plastic wrap as a guide. If the dough tears just press back together and proceed, it doesn't have to be perfect. Wrap each one in plastic wrap and freeze 4 hours (up to 1 month).
Preheat oven to 350°F. Line baking sheets with parchment paper. Slice dough into 1/4-inch-thick pieces and place 2 inches apart on baking sheets.
Bake 6 - 7 minutes or until puffed and set in the middle; don't over bake or cookies will become too crispy. Cool cookies on baking sheets 5 minutes then remove to wire racks and cool completely.
Meanwhile make the frosting: Beat together the butter and cream cheese at medium speed until creamy. On low speed gradually add powdered sugar beating until blended. Increase speed to medium and gradually add milk and peppermint extract beat until smooth and fluffy.
Place the frosting into a piping bag fitted with a small tip or a heavy-duty zip-top plastic freezer bag. Snip 1 corner of bag to make a small hole. Pipe about 2 teaspoons frosting onto half of cookies; top with remaining cookies and gently press to form a sandwich.
These cookies are delicious eaten at room temperature, or chill them 30 minutes. Store in a cool, dry place.