Line an 8-inch square pan with parchment, leaving overhang.
Melt semi-sweet chips, half the condensed milk, and 1 tbsp butter over low heat or in short microwave bursts.
Stir in vanilla and a pinch of salt until glossy.
Spread the chocolate layer evenly in the pan and chill 10 minutes to slightly set.
Melt white chips with remaining condensed milk and 1 tbsp butter until smooth.
Stir in peppermint extract and fold in half of the crushed candy canes.
Spread the white layer over the chocolate. Sprinkle on remaining candy canes and press lightly.
Chill until firm, about 2 hours. Lift out and cut into small squares.