Line your 8-inch or 9-inch square baking pan with foil or parchment with an overhang so you can lift the bark out after setting. Get the candy canes crushed and have your chocolate chips, sweetened condensed milk, marshmallow creme, and extracts ready.
In the saucepan over medium heat, combine the semi-sweet chocolate chips, half the sweetened condensed milk, and marshmallow creme. Stir constantly until the chocolate melts and the mixture is smooth. Remove from heat, stir in vanilla and peppermint extracts. Spread this layer evenly into the prepared pan and pop it into the fridge for a few minutes while you make the next layer.
Using the same saucepan (cleaned), combine the white chocolate chips, the other half of the sweetened condensed milk, and the marshmallow creme. Melt and stir until smooth. Remove from heat, stir in vanilla, peppermint extract, and the crushed candy canes. Pour this over the chilled dark chocolate layer, spread evenly, then sprinkle the extra candy-cane pieces on top for garnish. Cover the pan and refrigerate until completely set (about 4 hours or overnight).
Once firm, lift the bark block out of the pan using the foil or parchment overhang. Peel off the foil, then cut into squares or broken shards. Serve immediately or store as described below.