Preheat the oven to 350 degrees F. Butter a 9-by-13-inch baking dish, then line with parchment paper, leaving an overhang on 2 sides; butter the parchment.
Beat 2 1/2 sticks butter, the granulated sugar and 1 teaspoon vanilla in a large bowl with a mixer on medium-high speed until light and fluffy, 3 to 5 minutes. Reduce the mixer speed to low; add the flour and beat until just incorporated.
Transfer the dough to the prepared baking dish and set a piece of plastic wrap directly on the surface. Press into an even layer, then peel off the plastic. Bake until golden brown, 30 to 35 minutes. Transfer to a rack to cool slightly, then remove from the pan using the parchment. Cut into triangles.
Make the caramel: Combine the remaining 1/2 stick butter, the brown sugar, corn syrup, heavy cream, espresso powder and kosher salt in a small saucepan over medium heat. Cook, stirring occasionally, until a candy thermometer registers 235 degrees F, about 6 minutes. Remove from the heat and stir in the remaining 1 teaspoon vanilla and the vinegar. Let cool 5 minutes, then drizzle over the shortbread. Sprinkle with sea salt.