Make the Cake: Stir together the 1 tablespoon melted butter and 1 tablespoon cocoa powder until fully dissolved. Using a pastry brush, coat the entire interior of a standard 12-cup Bundt pan; set aside.
Adjust oven rack to lower-middle position and preheat oven to 350 degrees F.
Stir together the cocoa powder, chopped chocolate and espresso powder in a medium heatproof bowl. Pour the boiling water over the mixture and whisk until everything is completely melted and smooth. Set aside to cool to room temperature, then whisk in the sour cream. In a separate medium bowl, whisk together the flour, baking soda and salt.
Using an electric mixer, beat the butter, sugar and vanilla on medium-high speed until light and fluffy, about 3 minutes. Reduce the speed to medium and add the eggs one at a time, beating well between each addition and scraping the sides of the bowl as needed. Reduce the speed to medium-low and add one-third of the flour mixture and half of the chocolate mixture and mix until just incorporated. Scrape the sides of the bowl and repeat with half of the remaining flour mixture and the rest of the chocolate mixture. Finally, add the last of the flour mixture and beat until just incorporated. Scrape the sides and bottom of the bowl and beat on medium-low for 30 seconds to thoroughly combine the batter.
Pour the batter into the prepared Bundt pan and bake until a skewer inserted into the center of the cake comes out with just a few crumbs attached, 45 to 50 minutes. Place the pan on a wire cooling rack for 10 minutes, then turn the cake out onto the cooling rack and cool to room temperature, about 3 hours.
Make the Icing: In a medium saucepan set over medium-high heat, bring the butter, cocoa powder and evaporated milk to a boil, whisking occasionally. Add the powdered sugar, vanilla extract and salt and whisk continuously until the mixture becomes smooth, about 30 seconds.
Immediately remove the pan from the heat and pour the icing over the top of the cake, allowing it to drip down the sides. (The icing will begin to harden quickly after it is removed from the heat, so it's important to work quickly with it.)
The cake can be stored in an airtight container or wrapped with plastic wrap at room temperature for up to 3 days.