For the doughnuts: Whisk the buttermilk, egg and vanilla together in a small bowl and set aside. Add the flour, cocoa powder, sugar and salt to a fine-mesh sieve and sift into the bowl of a stand mixer fitted with the paddle attachment. With the machine running on low speed, slowly begin drizzling in the melted butter until the mixture looks like a fine-textured shortbread dough. Turn off the mixer and add the buttermilk mixture to the dry ingredients, then mix on low speed just until the dough starts to ride the paddle and comes together into a ball (add a little extra flour if needed).
Line a baking sheet with parchment paper and set aside. Generously coat the work surface with flour and turn the dough out onto the floured surface. Dust the top of the dough with more flour, then roll the dough into a 3/4-inch thick sheet. Use a 3-inch doughnut cutter to stamp out circles from the dough. Place the circles on the prepared baking sheet and repeat rolling and cutting with the scraps; you should get about 10 donuts.
Line a rimmed baking sheet with paper towels. Pour the oil into a large pot set over medium-high heat; the oil should fill the pot by about 3 inches-if it doesn't, add more. Heat the oil until it reads 365 degrees F on an instant-read thermometer. Add a few doughnuts to the hot oil, taking care not to overcrowd the saucepan (otherwise the doughnuts will stick together and not cook evenly). Cook until the doughnuts are golden brown, about 45 seconds on each side. Transfer to the paper towel-lined baking sheet to drain and cool. Repeat with the remaining dough circles, making sure the temperature of the oil returns to 365 degrees F between batches.
For the glaze: In a medium bowl whisk together the confectioners' sugar, milk and mint extract until smooth. Pour about one-quarter of the glaze into another bowl, cover with plastic wrap and set aside. Add a few drops of green food coloring to the remaining glaze in the bowl and whisk to combine.
For the ganache: In a small saucepan heat the cream over medium-high heat until it comes to a boil. Turn off the heat and add the chocolate chips and mint extract to the saucepan, whisking the mixture until smooth.
To finish the doughnuts: To a large bowl add the green glaze, the white glaze and a few drops of the chocolate-mint ganache. Do not mix. Dip the smoothest side of the cooled doughnuts into the bowl, then gently remove and place glazed-side up on a wire rack to set before serving. Repeat with the remaining doughnuts.
When re-rolling dough scraps, let the stamped-out doughnuts rest for 5 minutes before frying to prevent shrinkage. For added texture, after glazing the doughnuts sprinkle them with chocolate-covered poprocks or chopped and toasted nuts.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.