Prepare your baking pan by lining an 8×8 inch pan with parchment paper, leaving overhang on the sides for easy removal. Preheat your oven to 350°F (175°C).
Make the brownie batter by melting butter and chocolate together in a heatproof bowl set over simmering water (or microwave in 30-second bursts), stirring until smooth. Remove from heat and let cool slightly.
Whisk sugar into the chocolate mixture until well combined, then add eggs one at a time, whisking well after each addition. Stir in vanilla extract.
Sift the dry ingredients (flour, cocoa powder, and salt) directly over the chocolate mixture and fold gently with a rubber spatula just until no flour streaks remain. Do not overmix.
Transfer the batter to the prepared pan, spreading it evenly into the corners with your spatula. Bake for 20 minutes—the brownies should be partly set but still slightly soft in the center.
Meanwhile, prepare the caramel by heating sugar in a small saucepan over medium heat. Watch carefully as the sugar melts and turns amber (about 4-5 minutes), swirling the pan occasionally but not stirring.
Remove caramel from heat once it reaches a deep amber color and carefully add butter (it will bubble vigorously). Whisk until butter is incorporated, then slowly add cream while continuing to whisk.
Return the caramel to low heat for 1 minute, stirring constantly until smooth. Stir in ½ teaspoon sea salt and set aside to cool slightly while brownies continue baking.
Remove partially baked brownies from the oven and pour the warm caramel over the top, spreading it evenly. Sprinkle with additional sea salt.
Return to the oven and bake for an additional 15-20 minutes, until a toothpick inserted in the center comes out with a few moist crumbs.
Cool completely in the pan on a wire rack for at least 2 hours before cutting. For cleanest cuts, refrigerate for 1 hour after cooling.
Lift brownies out of the pan using the parchment paper overhang and cut into 16 squares with a sharp knife, wiping the blade between cuts.