Bake the cake:
Position a rack in the lower third of the oven and heat the oven to 350°F. Line the bottoms of three 9-inch cake pans with parchment and lightly grease the sides. Combine the flour, baking soda, and salt. Sift these together three times and set aside. In a mixing bowl, pour the boiling water over the cocoa and stir to blend. Refrigerate to cool to lukewarm, stirring occasionally to speed cooling. Stir in the cold water, yogurt, and vanilla. If necessary, refrigerate again to cool to room temperature before continuing.
In the bowl of an electric mixer, beat the butter and sugar until light in color and texture, 6 to 7 minutes at high speed with a hand-held mixer. If you’re using a heavy-duty mixer, use the paddle attachment at medium speed (the whisk attachment will aerate the batter too much). The butter and sugar mixture will remain somewhat granular; this is fine. Whisk the eggs briefly and dribble them slowly into the butter mixture, 2 to 3 minutes, stopping as needed to scrape down the bowl and beaters.
Stop the mixer and spoon one-third of the flour mixture into the mixing bowl. Beat on low speed, scraping the bowl at least once, just until all traces of flour are incorporated. Stop the mixer and pour in half of the cocoa mixture. Beat on low to medium speed, scraping the bowl at least once, just until the mixture is blended. Stop mixing and spoon half of the remaining flour into the bowl. Beat as before. Stop mixing to add the remaining cocoa mixture and beat as before. Add the last of the flour mixture and beat it in. Divide the batter evenly among the prepared cake pans, spreading the batter to level it.
Bake, rotating the pans halfway through, until the cake just begins to shrink from the sides of the pan and a toothpick inserted in the center comes out clean, 30 to 35 minutes. Let the cakes cool on a rack for about 5 minutes. Invert the pans to unmold. Peel off the parchment liners and turn the layers right side up on the rack. Let cool completely before filling and frosting.
Make the frosting:
Put the chocolate, butter, espresso powder, and salt in a large bowl. Bring the cream to a boil and pour it over the chocolate mixture. Stir until the chocolate is completely melted and smooth. Stir in the vanilla. Refrigerate until the mixture is cold and feels quite firm when you touch it, at least 2 hours. When you’re ready to frost the cake, beat the frosting with a hand-held electric mixer (it will seem a bit firm to beat at first), until the frosting lightens in color, has a spreadable but not-too-stiff consistency, and holds a nice shape. Frost the cake immediately, using about 2/3 cup between each layer and the rest for the top and sides.