In the bowl of a stand mixer fitted with the whisk attachment, or in a large mixing bowl using a handheld mixer, mix the softened butter, light corn syrup, and peppermint extract together until fully combined.
Add the powdered sugar and continue mixing on low speed until the mixture starts to come together, this will take a couple of minutes. The mixture will be very crumbly at first. Once the mixture starts to come together but is still a little crumbly, stop mixing and use your hands to work the dough together.
Sprinkle a piece of parchment paper with powdered sugar and turn the dough out onto the piece of parchment paper. Knead the dough together into a smooth ball, then flatten it out. Place another piece of parchment paper on top and use a rolling pin to roll the dough to ¼-inch thick.
Line a large baking sheet with parchment paper. Using a 1.25-1.5 inch circular cutter, cut out pieces of dough, and place onto the prepared baking sheet (if the dough is too soft to work with, place on a baking sheet and freeze for 5-10 minutes). Re-roll scraps of dough and continue cutting out circles until all the dough has been used.
Place the baking sheets with the cut-out peppermint patties in the freezer for 15-20 minutes to harden.
Once the patties are almost ready, add the chopped chocolate to a microwave-safe bowl. Microwave in 20-30 second increments, stirring well after each increment, until completely melted and smooth.
Remove the baking sheet from the freezer. Use a fork to dip each patty into the chocolate, tap off any excess chocolate in the bowl, and carefully place back on the baking sheet (I use a toothpick to slide it off of the fork). Repeat with the remaining patties until all of them are coated. If the patties become too soft to dip, place them back in the freezer for 10-15 minutes.
Once all of the peppermint patties are coated in chocolate, place the baking sheet in the refrigerator for 30-45 minutes to allow the chocolate to harden.
Serve and enjoy or store in an airtight container in the refrigerator.