For the cookies: Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Beat the butter, granulated sugar, brown sugar and salt in a stand mixer on medium speed until light and fluffy, about 2 minutes. Add the eggs 1 at a time, beating well in between each addition. Add the yolk and vanilla and beat until combined. Reduce the speed to low. Stir together the flour and cocoa powder in a small bowl, then add it to the dough in two batches, mixing well in between each.
Remove the dough from the bowl and place on a piece of plastic wrap (the dough will feel a bit wet but will come together once chilled). Form the dough into a disc and wrap tightly. Refrigerate until chilled, about 1 hour.
Place some cocoa powder in a small bowl. Scoop a 2-tablespoon piece of dough and roll into a ball. Dip the bottom of a flat-bottomed jar or glass in the cocoa powder and press the cookie out into a flat 3-inch disc. Place on the prepared baking sheet and repeat with the remaining dough. Bake the cookies until set and just starting to turn darker brown on the bottom, 9 to 11 minutes. Remove from the oven and let sit for 2 minutes, then transfer to a wire rack to cool completely, about 1 hour.
Meanwhile, make the buttercream: Melt the butter with the mint in a small saucepan over medium-low heat. Reduce the heat to low and continue to cook the butter, gently mashing the mint, until the butter starts to take on the flavor of the mint, about 15 minutes. Strain the butter into a small bowl and immediately place it in the fridge to solidify, about 30 minutes.
Scrape the butter into the bowl of a stand mixer and beat on medium-high speed until it is light and fluffy, about 5 minutes. Reduce the speed to low and beat in the confectioners' sugar in batches. Beat in the heavy cream, salt and food coloring. Transfer to a piping bag fitted with a round tip.
Build the cookies: Pipe buttercream on the bottoms of 9 cookies to cover in an even layer. Top each with another cookie, then chill until the buttercream is hardened, about 30 minutes.
Meanwhile, line a plate with parchment paper. Melt the chocolate and vegetable oil together in a small saucepan over medium heat until smooth. Working with one cookie at a time, gently place in the chocolate and spoon more chocolate over the top until completely coated. Use a slotted spatula or spoon to remove the cookie from the chocolate and place on the lined plate. Repeat with the remaining cookies. Refrigerate until the chocolate is hardened, about 10 minutes.