Dissolve espresso powder in a bowl with 2 tablespoons hot water.
Preheat oven to 350 degrees F (175 degrees C). Grease two 8-inch cake pans.
Beat white sugar, 1/2 cup butter, and brown sugar together in a bowl using an electric mixer until light and fluffy, about 3 minutes. Add eggs 1 at a time, blending well after each. Reduce mixer speed to low. Beginning and ending with flour, alternate adding flour and buttermilk in 5 rounds. Blend 2 teaspoons dissolved espresso into the batter.
Dissolve baking soda in the vinegar in a bowl; add to the batter. Mix until incorporated. Divide the batter equally between the prepared pans.
Bake in the preheated oven until golden brown, 32 to 35 minutes. Cool in the pans on a wire rack for at least 20 minutes before removing from pans and cooling completely, about 40 minutes more.
Place 1 1/2 plus 2 tablespoons butter in the bowl of an stand mixer with a whisk attachment. Whip on medium-high speed until smooth. Add confectioners' sugar and mix on medium-low until combined. Add salt, whipping on medium-high until smooth. Mix in heavy cream and vanilla extract on low until incorporated. Whip on high until frosting is light and fluffy, scraping down the bowl as needed.
Divide frosting into 5 equal portions. Leave 2/5 (2 portions) plain. Add remaining 2 teaspoons espresso mixture to another 2/5 (2 portions); stir until well blended. Add 2 tablespoons of caramel topping to the remaining 1/5 (1 portion); stir until well blended.
Level both cakes and place 1 on a platter. Spread 3 tablespoons of the remaining caramel sauce over it and top with a layer of espresso frosting. Place other cake on top. Spread a thin layer of vanilla frosting on the cake to seal in any crumbs; chill for 30 minutes.
Frost cake again with a thin layer of espresso frosting. Pipe vanilla frosting around the top edge. Fill the center with caramel frosting. Drizzle generously with the remaining caramel sauce.