Preheat oven to 350°F. Line a 9x13 inch pan with parchment paper or greased foil.
In a food processor, pulse the cookies (do not remove filling) until they're fine crumbs. You can also place cookies in a ziplock bag and use a rolling pin to crush cookies. Add melted butter and stir until combined.
Press cookie crumb mixture into the prepared pan. Bake for 10-12 minutes. Let cool completely.
With an electric mixer, beat softened butter, powdered sugar, peppermint extract, and milk until smooth and creamy, about 2-3 minutes.
Spread on top of cooled cookie crust. Freeze for 15 minutes.
In a heatproof bowl set over a pot of simmering water, melt chocolate and butter just until no lumps remain, about 5 minutes. Stir until smooth.
Spread chocolate over peppermint layer.
Freeze for 15 minutes or chill in the fridge for 30 minutes before slicing.
For clean slices, dip a sharp knife into hot water and wipe dry in between each slice.