Line a baking sheet with parchment paper or a silicone baking mat. Clear space in your fridge or freezer for chilling the patties later.

In a medium mixing bowl, beat the softened butter with the sweetened condensed milk using a hand mixer on medium speed until smooth and creamy, 1–2 minutes. You can also mix by hand with a sturdy whisk or wooden spoon.

Add the peppermint extract, vanilla extract, and fine sea salt. Beat just until combined. Taste and adjust the peppermint extract if you prefer a stronger mint flavor.

Add the powdered sugar 1 cup at a time, mixing on low after each addition. The mixture will start out soft and sticky and then become a thick, fondant-like dough that you can press with your fingers.

If the dough is too sticky to handle, add 1–2 tablespoons more powdered sugar at a time. If it is too dry and cracks when pressed, add a teaspoon of sweetened condensed milk or a tiny splash of milk until it reaches a soft, play-dough-like consistency.

Scoop about 1 tablespoon of the mint dough at a time and roll into a ball between your palms. Place on the prepared baking sheet and gently flatten into a disk about 1 1/2 inches wide and 1/4 inch thick.

Place the baking sheet in the freezer for 20–30 minutes, or in the fridge for about 45 minutes, until the patties are firm and easy to pick up. This helps them hold together when dipping.

Add the dark chocolate and coconut oil or shortening (if using) to a microwave-safe bowl. Microwave in 20–30 second bursts, stirring after each, until smooth and fully melted, about 1–1 1/2 minutes total. Alternatively, melt in a heatproof bowl set over a pot of barely simmering water.

Working with a few chilled peppermint centers at a time (keep the rest in the fridge or freezer), place one patty on a fork and lower it into the melted chocolate. Spoon chocolate over the top, then lift and gently tap the fork on the side of the bowl to let excess chocolate drip off. Slide the coated patty back onto the lined baking sheet using a toothpick or another fork.

While the chocolate is still wet, sprinkle the tops with crushed candy canes, flaky sea salt, sanding sugar, or leave plain. For a white chocolate drizzle, let the dark chocolate set first, then drizzle melted white chocolate over the patties in thin zigzags.

Chill the tray in the fridge for 20–25 minutes, or until the chocolate is firm and set. You can also let them set at room temperature if your kitchen is cool, though the fridge gives a snappier shell.

If any chocolate has pooled around the patties, trim the edges with a small knife for a neater look. Arrange on a platter, pack into tins, or store in the fridge or freezer until ready to serve.