Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans and line the bottoms with parchment paper.

In a large mixing bowl, whisk together the flour, sugar, cocoa powder, baking soda, and salt.

In another bowl, mix the egg, buttermilk, vegetable oil, and vanilla extract. Add the hot espresso and stir until combined.

Gradually add the wet ingredients to the dry ingredients, mixing until just combined.

Divide the batter evenly between the prepared pans and bake for 25 minutes or until a toothpick inserted in the center comes out clean.

Allow the cakes to cool in the pans for 10 minutes, then turn out onto wire racks to cool completely.

In a saucepan, melt the butter over medium heat. Continue cooking, stirring frequently, until the butter turns golden brown and has a nutty aroma.

Remove from heat and let cool slightly. In a mixing bowl, combine the browned butter with powdered sugar, milk, and vanilla extract. Beat until smooth and creamy.

Once the cakes are completely cool, place one layer on a serving plate. Spread a layer of frosting on top.

Top with the second cake layer and frost the top and sides of the cake with the remaining frosting.

Slice and serve, enjoying the rich flavors of chocolate and espresso!