Whisk the powdered sugar and cocoa together in a bowl to break up any lumps; set aside.
Melt the chopped chocolate in a heatproof bowl set over barely simmering water (or in the microwave at 60% power in 20–30 second bursts), stirring until smooth. Let it cool until just lukewarm to the touch.
In a stand mixer with the paddle attachment, beat the softened butter with the salt and espresso powder on medium speed until creamy and pale, 3–4 minutes, scraping the bowl once.
Blend in the vanilla, then pour in the cooled melted chocolate. Mix on medium speed until glossy and fully combined, about 1 minute.
On low speed, add the sugar-cocoa mixture in three additions, alternating with the heavy cream (start with 2 tablespoons total), mixing just until incorporated after each addition. Scrape down the bowl as needed.
Switch to the whisk attachment and whip on medium-high for 2 minutes to aerate. Adjust consistency: add up to 1 more tablespoon cream for silkier spread, or chill 10 minutes if too soft. Use immediately for spreading or piping.