For the brownies: Preheat the oven to 350 degrees F. Line an 8-inch square baking pan with parchment.
Melt the butter and chocolate together in a shallow heatproof bowl set over a saucepan with 1 or 2 inches of simmering water. Stir until completely melted and smooth.
Transfer the melted chocolate mixture to a medium bowl and stir in the sugar. Let cool, then whisk in the eggs, one at a time. Stir in the flour, espresso powder, salt and vanilla, and pour into the prepared baking pan.
Bake until a toothpick inserted in the center comes out clean, 20 to 25 minutes. Let cool completely.
For the caramel: While the brownies are baking, heat the sugar and 1/3 cup water in a heavy-bottomed medium saucepan over medium-high heat. Bring to a boil, swirling the pan but not stirring; cook until the mixture turns a golden amber color. Remove from the heat and immediately add the butter, swirling to combine; take great care to keep the pan at a distance when adding the butter, as the caramel will bubble up and foam. When the bubbling has subsided and the butter and caramel are combined, add the cream and vanilla and stir to incorporate. Set aside to cool.
Lift the cooled brownies out of the pan, cut into squares and place on a serving platter. Drizzle with the caramel and sprinkle with the sea salt. Serve with ice cream.