Stir bittersweet chocolate in a saucepan on medium-low heat until melted. Add peppermint extract and 2tsp vegetable oil. Cool to lukewarm (about 5 mins). Pour onto foil and spread evenly using a spatula or some kind of flat kitchen tool. Refrigerate until cold-at least 25min. Score the chocolate with a fork halfway through the chilling time. Remove any excess bits of dark chocolate.
Stir white chocolate in a metal bowl over a saucepan of simmmering water (do NOT let the water touch the bottom of the bowl) until melted. Add 2 tsp vegetable oil. Remove from heat and cool to 110*F (if you don't have a candy thermometer, this takes about 10mins or so-just make sure it's not very hot anymore but still liquidy. it needs to be only a little bit warm). Pour the white chocolate over the bittersweet chocolate and tip the balking sheet so that the white choc covers all of the dark choc. Don't use anything to spread it as it will mix up the chocolates. Sprinkle with the candy cane bits and press these lightly into the white choc. Chill to just firm-about 20mins.
Lift foil and chocolate onto counter or chopping board. Cut into 2inch strips. Cut each strip into 3 or 4 rectangles. Cut each rectangle into 2 triangles. (obviously you can make them whatever shape you want!)Chocolate must be kept in the refrigerator!