Combine sweetened condensed milk, peppermint extract, and vanilla in a large mixing bowl.
Add powdered sugar, 1 cup at a time, and mix with a Danish dough whisk or electric mixer. The dough is too stiff to use a wooden spoon or rubber spatula. A stand mixer is a great option as well.
Mix until a stiff dough forms. You should be able to form the dough into a ball without it sticking to your hands.
Lightly sprinkle your work surface with powdered sugar. Roll out peppermint patty dough 1/4" thick and cut using a small round cookie cutter. Place the rounds on wax or parchment paper-lined baking sheets.Note: I used a 1 7/8" (2") round biscuit cutter for mine. The smaller the cutter, the easier they'll be to dip later on so you can use a 1", 1 1/2", or 2". Don't have a cutter? No worries! You can roll the dough into small balls and press them flat with your fingers. Press scraps together and keep cutting out peppermint rounds until you've used it all up. Keep dusting the work surface with powdered sugar to prevent the dough from sticking.
Freeze rounds until firm, about an hour.
Melt chocolate chips and coconut oil in a microwave-safe bowl for 45-60 seconds. Stir. Continue to heat in 15-second increments until the chocolate chips are mostly melted. Remove from the microwave and stir until everything is melted and smooth.Note: if you overheat the chocolate it may cause the chocolate to turn slightly white when it hardens. So only heat until it is mostly melted and they'll continue to melt as you stir. Only pull out one sheet of peppermint rounds from the freezer at a time. You want to keep them as cold as possible! Use a fork to dip each peppermint patty round into the melted chocolate. Flip over, and tap the fork against the side of the bowl to allow excess chocolate to drip off. Scrape the fork against the edge of the bowl then use a second fork to help slide each dipped round back onto the parchment-lined baking sheet.Note: Even though you froze the peppermint rounds they will still be slightly soft when dipped into the chocolate. That's normal. Just work quickly. You'll have extra chocolate after dipping the patties. Spoon the excess into a ziplock bag and make a small cut in the corner. Drizzle the melted chocolate over the peppermint patties, if desired.
Allow patties to sit on the baking sheets until the chocolate has hardened, about an hour. The chocolate on the patties will remain soft for a couple of hours. You can definitely eat them and pick them up, but the chocolate may melt slightly on your fingers. After a couple of hours, they are much more stable and easy to pick up.
Patties can be stored at room temperature or in the refrigerator. We like storing ours in the refrigerator and eating them chilled.