Sift the powdered sugar into a large mixing bowl to remove any lumps. Add the softened butter and beat with an electric mixer on medium speed until the mixture becomes crumbly.
Pour in the heavy cream, peppermint extract, vanilla extract, and salt. Mix on low speed initially, then increase to medium and beat for 2-3 minutes until the mixture comes together into a smooth, thick dough. Taste and add more peppermint extract if desired.
Line two baking sheets with parchment paper or silicone baking mats. Using a small cookie scoop or tablespoon, portion out balls of filling (about 1 tablespoon each) and roll each portion between your palms to form a smooth ball.
Place the balls on the prepared baking sheets, spacing them about 1 inch apart. Gently press each ball down with the bottom of a glass or your palm to flatten into discs about 1/4 inch thick. Freeze for at least 1 hour, or until firm.
Set up a double boiler by placing a heatproof bowl over a pot of simmering water, making sure the bottom of the bowl doesn't touch the water. Add the chocolate and coconut oil (if using) and stir constantly until completely melted and smooth.
Remove from heat and let cool slightly, stirring occasionally, until the chocolate reaches about 90°F (it should feel barely warm to the touch). Alternatively, melt in the microwave in 30-second intervals, stirring between each interval.
Remove the frozen patties from the freezer, working with one baking sheet at a time. Using a fork, place one patty on the fork and lower it into the melted chocolate. Use a spoon to pour chocolate over the top, ensuring complete coverage.
Lift the patty out with the fork and gently tap the fork on the edge of the bowl to remove excess chocolate. Slide the coated patty onto a clean parchment-lined baking sheet using a toothpick or knife to help release it from the fork. Repeat with remaining patties, reheating the chocolate gently if it becomes too thick.
Let the chocolate-coated patties sit at room temperature for about 30 minutes until the chocolate is fully set. For faster setting, place them in the refrigerator for 15 minutes.
Once set, use a small knife to clean up any chocolate "feet" around the base if desired. Store in an airtight container with parchment paper between layers.