In a large bowl, with an electric mixer on high speed, beat butter and sugar until smooth; beat in egg yolks.
In another bowl, mix flour, baking powder, and salt. In a small bowl, combine milk, vanilla, and peppermint extract. Stir flour mixture into butter mixture alternately with milk mixture, blending thoroughly after each addition.
Divide dough in half. Leave one half in the mixing bowl and stir in a few drops of red food coloring.
Divide each half of dough in half again (four portions total); wrap dough portions in waxed paper or cooking parchment and chill until firm but still pliable, about 1 hour.
Unwrap dough and roll each of the four pieces into a smooth rope about 15 inches long. Twist each white rope with a red rope into a 13-inch-long combined rope (for a total of two ropes). Wrap twisted rolls in waxed paper; chill until firm, at least 4 hours, or up to 1 week (see notes).
Unwrap dough. Using a sharp knife, cut rolls into 1/8-inch-thick slices; place slices about 1 inch apart on buttered 12- by 15-inch baking sheets.
Bake cookies in a 350° oven until light golden, about 10 minutes; transfer to racks to cool completely.