Line an 8x8-inch baking pan with parchment paper, leaving an overhang for easy removal.
In a medium heatproof bowl, combine semi-sweet chocolate chips, 1/2 can sweetened condensed milk, and 1 tablespoon butter.
Melt the mixture over a double boiler (or in 30-second microwave intervals), stirring until smooth.
Pour the melted semi-sweet chocolate mixture into the prepared pan and spread it evenly. Refrigerate for 15 minutes, allowing it to set slightly.
In a separate heatproof bowl, combine white chocolate chips, the remaining 1/2 can sweetened condensed milk, and 1 tablespoon butter.
Melt the white chocolate mixture over a double boiler (or in 30-second microwave intervals), stirring until smooth.
Stir peppermint extract into the white chocolate mixture and fold in 1/4 cup of crushed peppermint candies.
Carefully pour the white chocolate layer over the semi-sweet chocolate layer, spreading it gently to avoid mixing.
Sprinkle extra crushed peppermint candies on top for decoration.
Refrigerate the fudge for at least 2 hours or until fully set.
Use the parchment paper to lift the fudge out of the pan.
Cut the fudge into squares with a sharp knife, cleaning the blade between cuts for clean edges.
Store in an airtight container at room temperature for up to one week or in the refrigerator for up to three weeks.