In the bowl of stand mixer, whisk together sugar, egg whites, salt, and cream of tartar. Place bowl over a pot of barely simmering water to create a double boiler. Ensure simmering water does not touch bottom of bowl.
Heat sugar-egg mixture until it reaches a temperature of 160°F, whisking often to distribute heat. Alternatively, if you don't have a thermometer, heat until the sugar has melted. Test this by rubbing mixture between fingers. The mixture should be smooth. You should not be able to feel the granules of sugar.
Return bowl to stand mixer. Attach whisk attachment to stand mixer. Beat medium at low-medium speed for 3 minutes until shiny. Increase speed to medium-high and continue beating to stiff peaks, about 8 minutes.
Add espresso powder and vanilla. Whip until thoroughly combined.
Transfer frosting to a piping bag fitted with a 16mm round tip (⅝-inch round tip or Ateco 808 round tip). Pipe a spiral of frosting onto each cupcake, leaving about ⅛-¼ inch border of exposed cupcake. Place frosted cupcakes on a sheet tray and keep chill in the fridge for 25-30 minutes until frosting is cold. Alternatively, place in the freezer for 10-15 minutes.