Line a 9-inch square baking dish with aluminum foil.

Combine chocolate chips and sweetened condensed milk in a saucepan over medium heat; cook, stirring frequently, until chocolate is melted, about 5 minutes. Remove from heat and immediately stir milk and butter into the chocolate mixture. Stir peppermint extract and vanilla extract into the mixture.

Spread chocolate mixture into the prepared baking dish. Refrigerate until set, about 2 hours.

Remove from dish by grabbing the foil and pulling straight up. Cut fudge into small squares. Store in covered container in refrigerator.