Mix sweetened condensed milk, peppermint extract, and vanilla in a stand mixer. Slowly add half the powdered sugar, then the rest, until a thick dough forms.
Dust parchment paper with powdered sugar, knead the dough briefly, and roll it into a ½-inch thick circle. Sprinkle more sugar on top to prevent sticking.
Cut out 1- to 1½-inch circles using a cookie cutter or small glass. If the dough softens, freeze for 15 minutes. Arrange patties on a parchment-lined sheet and freeze again for 15 minutes.
Melt dark chocolate in the microwave in 30-second bursts, stirring until smooth. Let it cool slightly so it coats better.
Dip each chilled patty in chocolate with a fork, coat both sides, and let the excess drip off. Place back on parchment, add sprinkles if desired, and refrigerate for 20 minutes until the chocolate sets.