Preheat the oven to 350°F (175°C). Grease and flour two small cake pans.

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

Add the egg, milk, oil, and vanilla extract to the dry ingredients and mix until well combined.

Stir in the boiling water until the batter is smooth.

Pour the batter evenly into the prepared cake pans.

Bake for 25 minutes or until a toothpick inserted into the center comes out clean.

Allow the cakes to cool in the pans for 10 minutes, then transfer to a cooling rack to cool completely.

In a mixing bowl, beat the softened butter until creamy.

Gradually add the powdered sugar and cocoa powder, mixing until combined.

Add the milk and vanilla extract, and beat until the frosting is smooth and fluffy.

Once the cakes are completely cooled, place one cake layer on a serving plate.

Spread a layer of frosting on top of the first cake layer.

Place the second cake layer on top and frost the top and sides of the cake.

Slice and serve the cakes.