Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Combine sugar, flour, cocoa, baking soda, baking powder, and salt in a medium bowl. Add eggs, milk, oil, vanilla extract, and peppermint extract; mix using an electric mixer until smooth. Whisk in boiling water. Pour batter into the prepared baking dish.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 30 to 35 minutes. Let cake cool 10 to 15 minutes before transferring to a wire rack to cool completely.
While cake is cooling, combine milk and peppermint extract in a small bowl and set aside. Cream butter in a separate bowl using an electric mixer until smooth. Pour in confectioners' sugar and peppermint-milk mixture alternately while continuing to beat. Add green food coloring until frosting reaches desired color. Whip frosting on high until fluffy, 2 to 3 minutes. Divide between two bowls. Stir chocolate chips into 1/2 the frosting using a spoon. Set mint-chocolate chip frosting aside.
Frost cooled cake with pure mint frosting without chocolate chips. Place cake in the refrigerator to set.
Meanwhile, combine chocolate squares and butter in a double boiler; stir until thoroughly melted. Remove from heat. Stir ganache vigorously to a smooth consistency.
Remove cake from the refrigerator. Cover entire cake with ganache; return to the refrigerator and allow to completely set, about 30 minutes.
Remove set cake from the refrigerator. Frost sides of cake with mint chocolate chip frosting.
