Place the chocolate in a glass measuring cup or microwave-safe bowl. Microwave at medium power for 2 minutes, then stir well. If needed, microwave for another 30 seconds at medium power and stir until completely melted. Set aside to cool.
Place the egg whites and sugar in the bowl of a stand mixer and whisk by hand just until combined.
Set up a double boiler: Fill a saucepan with a few inches of water and bring it to a simmer. Set the mixer bowl over the saucepan, making sure the bottom of the bowl does not touch the water.
Cook the egg white mixture over the simmering water, whisking occasionally, until it reaches 160°F on a candy thermometer. It should feel very warm and the sugar should be fully dissolved.
Once the mixture reaches 160°F, carefully return the bowl to the stand mixer. Using the whisk attachment, beat on high speed for about 8 minutes. If the bowl still feels warm, continue beating until it has cooled to room temperature. The meringue should be glossy and hold medium-stiff peaks. Switch to the paddle attachment.
With the mixer on medium-low, add the butter a few pieces at a time. Scrape down the bowl as needed. The mixture may look curdled at one point, but keep mixing and it will come together.If the buttercream looks too soft or loose at this stage, chill it briefly, then beat again until smooth. In a small bowl, stir together the espresso powder and boiling water until dissolved. Add the espresso mixture and vanilla to the buttercream and mix on medium speed for about 20 seconds. Reduce the speed to low and drizzle in the cooled melted chocolate.
Increase the speed to medium and beat until the buttercream is silky smooth, about 2 minutes more.