Line two (13x18-inch) baking sheets with wax paper or parchment paper.
In a medium mixing bowl, combine powdered sugar, sweetened condensed milk, peppermint extract, and vanilla extract. Beat on medium speed until a thick, smooth dough forms.
To shape the patties, choose one of the following methods to create about 48 patties:
Method 1 (By Hand): Scoop dough by tablespoonfuls. Roll into balls, then flatten into 1 ½-inch discs. Place on the prepared baking sheet.
Method 2 (Rolled and Cut): Roll dough out between two sheets of wax paper or parchment paper to about 1/4-inch thickness. Use a round cookie cutter (1 ½ to 2 inches) to cut out discs. Transfer to the prepared baking sheet.
Chill the patties in the refrigerator for 30 minutes, or until firm.
In a microwave-safe bowl or double boiler, melt 20 ounces chocolate chips or melting wafers with 1 tablespoon coconut oil (if using), stirring until smooth.
Dip each peppermint disc into the melted chocolate using a fork or dipping tool. Let excess chocolate drip off, then return to the lined baking sheet.
Refrigerate for 15 to 20 minutes, or until chocolate is completely set.
Serve chilled or at room temperature. Store in an airtight container in the refrigerator for up to 2 weeks.