Start by putting the sweetened condensed milk, peppermint extract, and vanilla extract into a mixing bowl. Add half of the powdered sugar and mix on low speed until combined.
Gradually add the rest of the powdered sugar and beat until you have a smooth, firm dough. Test by pinching a piece and rolling it into a ball—it should hold its shape. If it’s too sticky, add a tablespoon of powdered sugar at a time. If it’s too dry, add a teaspoon of water at a time.
Sprinkle some powdered sugar on a piece of parchment paper. Place your dough on top, sprinkle a little sugar over it, and roll it out into a ½-inch thick circle.
Lightly coat a 1–1.5 inch round cookie cutter with cooking spray and cut out circles from the dough. If the dough is too soft to cut cleanly, pop it in the freezer for a few minutes.
Transfer the peppermint rounds to a parchment-lined baking sheet and freeze for 20 minutes.
While the patties chill, melt the chocolate candy melts in a microwave-safe bowl, heating in 30-second intervals and stirring between each until smooth.
Using a fork, dip each peppermint patty into the melted chocolate, place it back on parchment, and immediately add holiday sprinkles. Let the chocolate set completely. Store your treats in the refrigerator until ready to serve.