In a mixing bowl, beat the shortening and butter for 30 seconds.
Add about half of the flour, the sugar, egg, milk, vanilla, baking soda, and salt to beaten mixture. Beat until thoroughly combined, scraping sides of bowl occasionally. Then, beat or stir in the remaining flour.
Divide dough in half. Set one portion aside. To the remaining portion, knead in the peppermint extract and the food coloring.
To shape dough, roll out each portion of dough between 2 sheets of waxed paper to form a 12x11-inch rectangle. Remove top sheet of waxed paper from plain and pink doughs. Invert plain dough on top of pink dough. Peel off top sheet of paper. From the long side, roll up, jelly-roll style, removing bottom sheet of paper as you roll. Cut roll in half, crosswise. Wrap and chill for 4 to 24 hours.
Cut dough into 1/4-inch-thick slices. Place 2 inches apart on an ungreased cookie sheet. Bake in a 375 degree F oven for 8 to 10 minutes or until edges are firm and bottoms are lightly browned. Remove cookies and cool on wire rack. Makes about 60 cookies.