Set out a large mixing bowl and a hand mixer. (Or use an electric stand mixer.) Add the sweetened condensed milk, peppermint extract, and vanilla extract to the bowl. Beat to combine. Then slowly add 1/2 cup of powdered sugar at a time until you have a thick pliable peppermint dough that can hold its shape.
Place a little extra powdered sugar on a clean work surface. Dump the dough onto the work surface and fold into a firm ball. Then roll the ball out into an even ¼-inch sheet.
Use a 1-inch cookie cutter to cut out peppermint discs, cutting close together. Line a baking sheet with parchment paper and move the discs to the baking sheet. You should be able to cut about 80 peppermint discs from the dough. You may need to gather and re-roll the scraps once or twice.
Once all the discs are cut, move the baking sheet to the freezer. Freeze the discs until they are very hard, at least one hour.
When ready to dip, place one bag of chocolate chips in a microwave safe bowl. Remove one handful of chips for later use. Microwave the chocolate in 30 second increments, stirring in between, until the chocolate chips are melted and smooth. Stir in the oil. Then throw the last handful of chocolate chips back into the chocolate and stir until smooth to temper the chocolate.
Pull 3-4 frozen peppermint discs out of the freezer at a time. Place a frozen disc on a fork, lower into the chocolate, coat on all sides, then tap vigorously on the side of the bowl to drain off excess chocolate. Place the dipped peppermint Patty on a piece of parchment paper and sprinkle with decorative candy sprinkles if desired.
Work in small batches so that the peppermint patties stay frozen while you are dipping. At some point you may need to reheat the chocolate.
When you’ve used all of the chocolate from the first bag, melt the second bag of chocolate chips and continue dipping.
Notes
You can store the homemade peppermint patties in the refrigerator for up to a month, or leave them out at room temperature for up to one week.