• Preheat the oven to 350°F. Butter the individual wells in the doughnut pan with the softened butter.

Make the Batter

• Place the cocoa and espresso powder in a medium-sized mixing bowl and whisk to combine. Add the boiling water and stir the ingredients together until the mixture forms a paste and with no visible cocoa streaks. Cover the bowl with plastic wrap and let the chocolate paste cool until it is barely warm.
• In a small bowl whisk together the flour, baking powder and salt. Set aside.
• Whisk the egg into the cooled chocolate paste until fully combined and the mixture is shiny. Next whisk in the brown sugar. Add the milk, oil, and vanilla extract and whisk in these ingredients until thoroughly combined. Let the mixture sit for 2 minutes to give the sugar time to dissolve.
• Add the dry ingredients to the wet ingredients and use a spatula to fold them together. Scrape the bottom frequently to ensure there is no unmixed flour. The batter will be slightly lumpy which is okay.

Bake the Doughnuts

• Transfer the batter into an 18-inch piping bag (see notes below). Pipe the batter into each doughnut well, about 65 grams each. Bake the doughnuts for 8-10 minutes until they are risen with small cracks on top. If you’re using an instant-read thermometer, the temperature will be 195-198°F (90-92°C)
• Remove the doughnuts from the oven. Wait for 2 minutes, then invert the pan onto a cooling rack to release the doughnuts. You make need to shake the pan a little if any of the doughnuts stick, but they should all come out. The doughnuts can be eaten warm, but if you’re going to add a chocolate glaze let the doughnuts cool to room temperature.

Add the Chocolate Glaze

• While the doughnuts are baking make the glaze. Place the butter and chopped chocolate in a microwave safe bowl. Microwave the ingredients for 15 second, then remove and stir the ingredients even if nothing appears melted. Continue to microwave in 15-second intervals stirring after each interval. When 2/3 of the chocolate is melted remove the bowl from the heat and stir together until the glaze is smooth and the ingredients are well combined.
• Let the chocolate glaze cool to about 90°F before dipping the doughnut in the glaze. To dip the doughnuts, hold a cooled doughnut by the edges and dip the side that was touching the pan (the bottom) into the glaze. This side has a slightly rougher texture, which helps the glaze cling better. Lift the doughnut straight up and let the excess glaze drip back into the bowl for a few seconds. Place the glazed side up on a wire rack set over parchment paper or a baking sheet. Once you dipped all the doughnuts, dip them a second time a nice thick and smooth glaze.
• Allow the glaze to set for about 10 minutes before serving.

Store the Doughnuts

• The doughnuts are best eaten within 2 days. Store them in an airtight container.