Preheat an oven to 350F. Line an 8'' square baking pan with parchment paper. For easy removal, leave enough overhang of the parchment to lift the sides when finished baking. Coat with butter or nonstick baking spray.
Place unsalted butter and semi-sweet chocolate in a medium saucepan set over medium heat. Cook until melted, stirring occasionally, then add cocoa powder and espresso powder. Whisk well.
Place eggs, sugar, and vanilla extract in the bowl of a stand mixer fitted with whisk attachment. Whisk until the mixture turns pale yellow and smooth, about 3 minutes.
Slowly add the melted chocolate mixture to the eggs, then whisk to combine. Dump in all-purpose flour and Kosher salt all at once, then whisk (or fold with a spatula) just until no flour streaks remain.
Transfer the brownie batter to the prepared pan, then bake. The brownies should have crisp, firm edges and just a slight jiggle in the center. Sprinkle immediately with a generous pinch or two of flaky sea salt.