In a large bowl, combine the powdered sugar, milk, peppermint extract, and melted butter.

Mix until a smooth dough forms. If the dough is too dry, add a little more milk, one teaspoon at a time. If it's too wet, add a little more powdered sugar.

Roll the dough out to about 1/4 inch thickness on a surface lightly dusted with powdered sugar. Use a round cookie cutter (about 1.5 inches) to cut out the peppermint patties.

Place the patties on a parchment-lined baking sheet and chill in the freezer for at least 15 minutes to firm up.

In a double boiler or microwave-safe bowl, melt the chocolate and shortening together, stirring until smooth.

Dip each chilled peppermint patty into the melted chocolate, coating completely. Use a fork to lift the patty out and tap off any excess chocolate.

Place the chocolate-covered patties back on the parchment-lined baking sheet and refrigerate for at least 30 minutes, or until the chocolate is set.