Start by preheating your oven to 350°F (175°C) and lining two baking sheets with parchment paper. In a large mixing bowl, use an electric mixer to cream together the butter and granulated sugar until the mixture is smooth and fluffy, which should take about 3–4 minutes. Add the vanilla extract and the egg, and continue mixing until everything is fully incorporated. Gradually add in the flour, cocoa powder, baking soda, and salt, mixing just until a soft dough forms. The dough should be cohesive and may feel slightly tacky but not overly sticky. Roll the dough into 16 balls, approximately 1¾ inches in diameter (a little larger or smaller is fine depending on your preference). Arrange them on the prepared baking sheets and bake for 8 to 10 minutes, just until the edges are set and the centers appear slightly glossy. Allow the cookies to cool on the baking sheets for 10 minutes before transferring them to a wire rack to cool completely.

In a medium bowl, beat together the powdered sugar, butter, milk, and mint extract until smooth and creamy. If using, add a drop or two of green food coloring to achieve the desired shade. Adjust the consistency by adding more powdered sugar if the frosting is too thin, or a splash more milk if it’s too thick. Once the cookies are completely cool, spread the frosting evenly over the tops. Place the frosted cookies in the freezer for at least 15 minutes to help the frosting firm up before adding the ganache.

Combine the chocolate and oil in a medium, microwave-safe bowl. Heat in the microwave in short bursts of 30 to 40 seconds, stirring thoroughly between each interval, until the chocolate is just melted and smooth. Remove the chilled cookies from the freezer and spoon the ganache over the frosting. The cold cookies will help control the flow of the ganache, keeping it from running off the edges. Once topped, the cookies can be served immediately or left at room temperature to thaw completely—the ganache will stay set and firm either way.