Sift the powdered sugar into a large mixing bowl to remove lumps.
Add the corn syrup, softened butter, and 4 to 5 drops of peppermint extract to the sifted powdered sugar.
Beat the mixture with an electric mixer (or vigorous whisking) until it forms a smooth, pliable dough.
Portion the mint dough into bite-sized balls, then flatten each slightly; place them on a silicone mat or waxed paper and refrigerate for about 20 minutes to firm up.
When ready to dip, press each chilled piece with the bottom of a small glass to form a patty.
Set up a double boiler (or a heatproof bowl over a saucepan of simmering water) and melt the dark chocolate with the vegetable shortening, stirring until smooth.
Using a fork or dipping tool, drop each peppermint patty into the melted chocolate, coat completely, flip to cover the bottom, then lift out and place on the prepared mat or paper to set.
Allow the chocolate to harden before storing the patties in an airtight container.