In a mixing bowl, combine the powdered sugar, sweetened condensed milk, peppermint extract, and cornstarch. Use an electric mixer or whisk to blend until the mixture is smooth and holds together.
Dust a clean surface with cornstarch. Place the mixture on the surface and roll it out to a thickness of about 1/4 inch (0.6 cm).
Use a cookie cutter or round cutter to cut out circles from the rolled mixture. Place the circles on a baking sheet lined with parchment paper. Chill in the refrigerator for about 15 minutes to firm up.
In a double boiler or microwave-safe bowl, melt the semi-sweet chocolate chips and vegetable shortening together until smooth, stirring frequently.
Remove the chilled peppermint circles from the refrigerator. Using a fork or dipping tool, dip each circle into the melted chocolate, ensuring it is fully coated, then lift it out and allow excess chocolate to drip off.
Place the dipped peppermint patties back on the parchment-lined baking sheet. Repeat until all are coated.
Chill the complete patties in the refrigerator for at least 30 minutes or until the chocolate has hardened.
Store the finished peppermint patties in an airtight container in the refrigerator until ready to serve.