Equipment: a candy thermometer; a stand mixer fitted with whisk attachment
How to make it
Bring 1 3/4 cups sugar and water to a boil in a 3-quart heavy saucepan over medium heat, stirring until sugar has dissolved, then wash down any sugar crystals from side of pan with a pastry brush dipped in cold water. Boil, without stirring, until it registers 220 to 225°F, 15 to 20 minutes.
At this point, while continuing to boil syrup, beat whites with espresso powder, vanilla, cream of tartar, and salt in mixer at medium speed until they just hold soft peaks. Add remaining 1/4 cup sugar, 1 tablespoon at a time, beating, and beat until whites just hold stiff peaks.
When syrup reaches soft-ball stage (238 to 242°F), immediately pour syrup in a slow stream down side of bowl into whites (avoid beaters) while beating at high speed. Beat until completely cool, 25 to 30 minutes. With mixer at medium speed, add butter 1 tablespoon at a time, beating well after each addition (see cooks’ note, below) and until buttercream is smooth. (Mixture may look curdled before all butter is added but will come together at end.)
Transfer 2 cups buttercream to a small bowl and stir in chocolate. If buttercreams are too soft to spread, chill, stirring occasionally