In a large bowl, beat the butter and sugar with an electric mixer for about 2 minutes, until light and fluffy.
Mix in the vanilla extract and milk until smooth.
In a separate bowl, whisk together the flour, cocoa powder, cornstarch, baking soda, and salt.
Gradually add the dry ingredients to the butter mixture, mixing on low speed just until a soft dough forms. If the dough seems crumbly, gently knead it by hand for 20–30 seconds. Add up to 1 teaspoon additional milk only if necessary.
Divide the dough into two equal portions and flatten each into a disc. Wrap tightly in plastic wrap and refrigerate for 1 hour.
Preheat the oven to 325°F (165°C). Line two baking sheets with parchment paper.
Let the chilled dough sit at room temperature for about 5 minutes if it feels very firm.
Roll the dough between two sheets of parchment paper until it is approximately 1/16 to ⅛ inch thick.
Cut out 1¾-inch circles and place them about 1 inch apart on the prepared baking sheets.
Bake for 10 to 12 minutes, or until the cookies look dry and set. Do not allow the edges to brown.
Cool the cookies on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely. They will become crisp as they cool.
Place the semisweet chocolate, dark chocolate, and coconut oil in a microwave-safe bowl. Microwave in 30-second intervals, stirring between each, until smooth. Stir in the peppermint extract.
Using a fork, dip each cooled cookie into the melted chocolate, coating it completely. Tap the fork gently against the side of the bowl to remove excess chocolate.
Place the coated cookies on a parchment-lined baking sheet.
Refrigerate for 15–20 minutes, or until the chocolate coating is firm.
Serve chilled or frozen for the most authentic Thin Mint experience.