Line an 8-inch square baking pan with parchment paper, leaving a 2-inch overhang on two sides. Crush the candy canes in a zip-top bag with a rolling pin until mostly fine, with a few chunky pieces left for texture.
Combine the chocolate chips, sweetened condensed milk, and butter in a medium saucepan over low heat. Stir constantly with a rubber spatula until melted, smooth, and glossy, 4 to 5 minutes; do not let it bubble.
Take the saucepan off the heat and stir in the peppermint extract until evenly combined.
Pour the fudge into the lined pan and spread it level. Scatter the crushed candy canes over the top and press them in gently.
Refrigerate uncovered for 2 hours, until the fudge is firm and lifts cleanly from the parchment.
Lift the block out by the parchment overhang and cut into 6 rows by 6 columns for 36 pieces. Store in an airtight container in the fridge for up to 1 week.
