FOR THE CAKE Preheat oven to 350ºF.
Grease and flour sides of a 9-inch by 2-inch round cake pan.
Place a round of parchment in the bottom of the pan; set aside.
Whisk the coffee, milk and vanilla extract together in a small bowl; set aside.
Whisk the flour, baking soda and salt together in a medium bowl; set aside.
In the bowl of your Stand Mixer, cream the butter and sugars until light and fluffy.
Add the eggs, one at a time, beating well after each addition.
On low speed, pour in the melted chocolate. Increase speed to high and beat until mixed well.
Add 1/3 of the flour mixture and beat well.
Add 1/3 of the milk mixture and beat well.
Repeat, alternating dry and wet ingredients, scraping the sides of the bowl and beating well after each addition.
Divide the batter evenly into prepared pans.
Bake for 20 to 25 minutes or until a toothpick inserted in center comes out clean.
Let cool in pan for 10 minutes, then invert onto a wire rack and cool completely before frosting.
FOR THE FROSTING Combine egg whites and salt in a large bowl.
Stir together brown sugar and water in a small heavy saucepan and bring to a boil over moderately high heat, washing down side of pan with a pastry brush dipped in water.
When sugar syrup reaches boil, start beating whites with an electric mixer at medium-high speed until frothy, then add lemon juice and beat at medium speed until whites just hold soft peaks.
(Do not beat again until sugar syrup is ready.)
Meanwhile, put candy thermometer into sugar syrup and continue boiling until syrup reaches 238–242°F.
Immediately remove from heat and pour into a heatproof 1-cup glass measure.
Slowly pour hot syrup in a thin stream down side of bowl into egg whites, beating constantly at high speed.
Beat meringue, scraping down bowl with a rubber spatula, until meringue is cool to the touch, about 6 minutes. (It’s important that meringue is properly cooled before proceeding.)
With mixer at medium speed, gradually add butter 1 piece at a time, beating well after each addition until incorporated.
(If meringue is too warm and buttercream looks soupy after some of butter is added, briefly chill bottom of bowl in a large bowl filled with ice water for a few seconds before continuing to beat in remaining butter.)
Continue beating until buttercream is smooth.
(Mixture may look curdled before all of butter is added, but will come back together before beating is finished.)
Add vanilla and beat 1 minute more.
