Heat the oven to 350°F with a rack in the center. Lightly grease a 9-inch square light-colored metal pan, then line it with two crossing parchment strips, leaving enough overhang to lift the brownies later. Set the pan on a heatproof surface. Warm the heavy cream in a heatproof jug until steaming but not boiling, about 20 to 30 seconds in a microwave, and keep it close to the stove for the caramel.
Place the caramel sugar in a small heavy-bottomed saucepan over medium heat. Cook without stirring, swirling the pan gently once the edges liquefy, until all the sugar has melted and turned a deep amber, 6 to 9 minutes. Remove the pan from the heat and gradually whisk in the warm cream, keeping your hands clear of the vigorous steam. Whisk in the caramel butter and fine sea salt until smooth. Transfer to a bowl and refrigerate for 25 to 30 minutes, stirring once, until thick but still spoonable.
Combine the brownie butter and chopped dark chocolate in a medium heatproof bowl. Microwave in 20-second intervals at half power, stirring thoroughly after each interval, until the mixture is almost melted. Continue stirring off the heat until completely smooth so residual heat finishes the job without scorching the chocolate. Let it stand for about 5 minutes; it should remain fluid but feel only slightly warm before it is combined with the eggs.
In a large bowl, whisk the brownie sugar, eggs, vanilla, and espresso powder for 60 to 90 seconds, stopping when the mixture is smooth, slightly thickened, and evenly blended but not pale or foamy. Slowly whisk in the warm chocolate mixture until glossy and uniform. Sift the flour, cocoa powder, and fine sea salt directly over the bowl. Fold with a silicone spatula, scraping the bottom and sides, just until no dry flour remains; prolonged mixing can make the cooled brownies firmer.
Spread about two-thirds of the brownie batter evenly in the prepared pan, reaching fully into the corners. Drop half of the thickened caramel over it in small spoonfuls. Cover with the remaining batter, then add the remaining caramel in spaced dollops. Drag a skewer through the layers only three or four times to create broad ribbons without blending them together. Sprinkle with flaky sea salt. Bake for 31 to 36 minutes, until the edges are set, the center does not slosh, and a toothpick finds moist crumbs but no glossy liquid batter.
Set the pan on a wire rack and cool the brownies completely in the pan for at least 2 hours; the center continues setting as the chocolate and caramel cool. Refrigerate the covered pan for 30 minutes to firm the layers before cutting. Lift the slab out with the parchment and peel the paper away. Use a large sharp knife to divide it into 16 squares, wiping and warming the blade under hot water, then drying it thoroughly, between cuts. Let the squares lose their refrigerator chill before serving.