A filling of custardy pastry cream is layered within the Mocha Bubble Cake. Ultra-smooth buttercream adds a rich finishing touch.
Make the mocha buttercream: Combine espresso powder and hot water in a small bowl; stir until espresso dissolves. Heat egg whites and sugar in a heatproof bowl of a mixer set over a pan of simmering water, whisking until whites are warm to the touch and sugar dissolves, 2 to 3 minutes. (Mixture should feel smooth when rubbed between 2 fingers.) Whisk egg-white mixture with a mixer on medium-high speed until stiff, glossy peaks form and mixture cools completely, about 10 minutes.
Reduce speed to medium-low. Add butter, a few tablespoons at a time, whisking well after each addition. Switch to the paddle attachment, and reduce speed to low. Beat in espresso mixture and melted chocolate until smooth, then beat to reduce air bubbles, 3 to 5 minutes. Cover with plastic wrap, and let stand at room temperature until ready to use.
Make the mocha pastry cream: Prepare an ice-water bath. Combine espresso powder and hot water in a small bowl; stir until espresso dissolves. Whisk eggs, egg yolks, and 1/4 cup sugar in medium bowl.
Bring remaining 1/4 cup sugar, the milk, vanilla seeds, cornstarch, and 1/4 teaspoon salt to a boil in a medium saucepan over medium-high heat, whisking constantly. Remove from heat, and gradually whisk one-third of hot-milk mixture into egg mixture. Whisk entire mixture back into saucepan. Raise heat to medium-high, and bring to a boil, whisking constantly. Cook, continuing to whisk, until mixture is thick enough to hold its shape when lifted with a spoon, about 3 minutes.
Remove from heat. Whisk in butter, 1 piece at a time. Whisk in espresso mixture and chopped chocolate until smooth. Strain mixture through a fine sieve into a bowl. Cover with parchment or plastic wrap, pressing it directly on surface to prevent a skin from forming. Transfer to ice-water bath; let stand until cooled, about 30 minutes. Refrigerate until chilled and firm, at least 2 hours.