Heat oven to 350 degrees F.
Line 4 sheet pans with parchment paper. Set aside.
Melt white chocolate in the microwave in 15 second bursts, stirring after each, until smooth. Set aside.
Combine flour and baking powder. Set aside.
Beat butter and sugar until light and fluffy.
Add egg and beat until well combined.
Stir in milk, vanilla. and peppermint oil. (Alternately, stir peppermint oil into half the dough when you divide it.)
Add flour mixture and stir just until mixed.
Stir in white chocolate.
Divide dough in half.
Add red color to half the dough (the peppermint half, if separated).
Sprinkle work surface with powdered sugar.
Pinch off a 1/2 ounce portion of white dough and roll into a 3-inch long log.
Repeat with red dough.
Gently twist the two doughs together.
Roll gently into a 7-inch striped log.
Gently lift with cookie spatula that has been sprinkled with powdered sugar.
Place on prepared pan and form into candy cane shape.
Bake for 7-8 minutes, until just starting to color.
Allow to cool completely on the pans.
Gently loosen and lift off with a cookie spatula.