Preheat the oven to 350°F (177°C) and place a rack in the lower-middle position.

Prepare the Bundt pan: stir together 1 tablespoon melted butter and 1 tablespoon cocoa powder until smooth. Using a pastry brush, coat the entire interior of a standard 12-cup Bundt pan with the mixture. Set the pan aside.

Make the chocolate base: in a medium heatproof bowl combine 3/4 cup (64.5 g) natural unsweetened cocoa powder, 6 ounces (170.1 g) finely chopped bittersweet chocolate, and 1 teaspoon instant espresso powder (optional). Pour 3/4 cup (187.5 ml) boiling water over the mixture and whisk until fully melted and smooth. Set the bowl aside to cool to room temperature.

When the chocolate mixture is at room temperature, whisk in 1 cup (230 ml) sour cream until smooth.

In a separate medium bowl whisk together 1 3/4 cups (218.75 g) all-purpose flour, 1 teaspoon baking soda, and 1 teaspoon salt. Set this dry mixture aside.

In the bowl of an electric mixer (or using a handheld mixer), beat 3/4 cup (170.25 g) unsalted butter, 2 cups (440 g) light brown sugar, and 1 tablespoon vanilla extract on medium-high speed until light and fluffy, about 3 minutes.

Reduce the mixer speed to medium and add the 5 eggs one at a time, beating well after each addition and scraping down the sides of the bowl as needed.

Prepare to combine: divide the dry flour mixture into three equal portions, and divide the cooled chocolate-sour cream mixture into two equal portions.

With the mixer on medium-low, add the first third of the flour mixture and mix just until beginning to incorporate. Add half of the chocolate mixture and mix until just combined. Scrape the sides of the bowl.

Add the second third of the flour mixture and mix until just combined. Add the remaining half of the chocolate mixture and mix until just combined. Scrape the bowl again.

Add the final third of the flour mixture and mix until just incorporated. Scrape the sides and bottom of the bowl and beat on medium-low for about 30 seconds to fully combine the batter without overmixing.

Pour the batter evenly into the prepared Bundt pan, smooth the top if needed, and bake on the lower-middle rack for 45 to 50 minutes, or until a skewer inserted into the center comes out with just a few moist crumbs attached.

Remove the pan from the oven and place it on a wire cooling rack. Let the cake cool in the pan for 10 minutes. Then invert the pan onto the rack to release the cake and cool the cake completely to room temperature, about 3 hours.

Make the icing: place 1/2 cup (113.5 g) unsalted butter, 1/2 cup (43 g) cocoa powder, and 6 tablespoons evaporated milk in a medium saucepan over medium-high heat. Whisk occasionally and bring the mixture to a boil.

Once boiling, add 3 3/4 cups (450 g) powdered sugar, 1 teaspoon vanilla extract, and 1/4 teaspoon salt. Whisk continuously until the icing is smooth, about 30 seconds. Immediately remove the pan from the heat.

Place the cooled cake on a wire rack set over a tray or baking sheet to catch drips. Pour the warm icing over the top of the cake, starting in the center and allowing it to drip down the sides. Work quickly—the icing begins to set as it cools.

Let the icing set. Store the cake in an airtight container or wrapped in plastic wrap at room temperature for up to 3 days.