To make the sprinkles: In the bowl of an electric mixer fitted with a paddle attachment (or using a handheld electric whisk), combine confectioner's sugar, egg white, extract and salt. Mix the ingredients on low speed until a paste forms. The paste should have the consistency of liquid glue. Scrape down the sides of the bowl and continue mixing until all the confectioner's sugar is fully incorporated. Add the food coloring and mix again until it's the perfect hue of mint.
Transfer the icing to a pastry bag,fitted with a medium pastry tip (you want the sprinkles to be large!). Pipe out long, thin lines on a parchment-lined baking sheet, ensuring that the lines do not touch each other. Allow the sprinkle lines to dry, uncovered, for 24 hours. Once the piped lines have dried completely, inside and out, use a bench scraper or a butter knife to break and/or chop the piped lines into 1/2-inch, LARGE sprinkle sized pieces. Use immediately (like on dis doughnut), or store in a dry, airtight container for up to 1 month. Just a heads up, I ended up being a bit psycho about making the sprinkles look as clean as possible. I took a knife and cleaned up the sprinkles so the edges were sharp and clean. You don't have to be this way or you can be--up to you!
To make the doughnut: Preheat your oven to 350 degrees F. Spray your doughnut pans with cooking spray and set aside. In a medium bowl, whisk together the flour, cocoa powder, baking powder, baking soda and salt.
In a medium saucepan, set over medium heat, add the butter. When melted, add the chocolate chips and espresso powder. Allow to stand and melt for a minute or two and then mix with a spatula until smooth. Turn the heat off and then stir in the sugar and egg yolk. Lastly, whisk in the buttermilk and vanilla extract.
In one batch, add the wet ingredients to the dry and mix just until you no longer see any speckles of flour. Fill the doughnut pans half way. Transfer the pans to the oven to bake for 15 to 20 minutes, until they've doubled and firm to the touch. Allow to cool in the doughnut pans for a minute or two and then run a butter knife along the outside. Invert the doughnuts onto a cooling rack. Allow to cool completely.
To make the frosting: In a medium bowl, whisk together the powdered sugar, cocoa powder, vanilla extract and 2 tablespoons of milk. Add more milk or water, one tablespoon at a time, until it reaches a thick yet pourable consistency.
To assemble: If you need to, you may have to level off the inside halves of the doughnuts. Take a serrated knife and lop off the excess cake. Place the halves on top of one another. Pour the glaze onto the top of the big doughnut and arrange large sprinkles on the glaze's surface. (I tried dipping the top layer into a shallow pan and it was scary and almost felt like the cake was going to break!) Slice it up and serve.
To make ahead: This is great to make ahead. Make the sprinkles ahead. Also, the cake keeps well wrapped tightly in plastic wrap for up to 3 days. It seriously is delicious still! Glaze can be made up to 1 day ahead--must whisk vigorously just before using.