In a medium saucepan, combine sugar and water over medium heat. Stir occasionally until the sugar dissolves, then allow it to cook without stirring until it turns a deep amber color, about 8-10 minutes. Remove from heat and whisk in the salted butter and heavy cream until smooth. Let the salted caramel cool to room temperature while you prepare the cake batter.

Preheat your oven to 350°F (175°C). Grease two 6-inch round cake pans with butter and line the bottoms with parchment paper. In a large bowl, combine all-purpose flour, granulated sugar, brown sugar, cocoa powder, baking powder, baking soda, and salt. In another bowl, whisk together eggs, Greek yogurt, buttermilk, melted butter, espresso powder, and vanilla extract. Gradually mix the wet ingredients into the dry, stirring until just combined.

Pour the cake batter evenly into the prepared pans, smoothing the tops with a spatula. Bake in the preheated oven for approximately 25 minutes or until a toothpick inserted into the center comes out clean. Keep an eye on them; if the edges start to pull away from the pan, they're ready. Remove the cakes and let them cool in the pans for 10 minutes before transferring to a wire rack.

Allow the cakes to cool completely on the wire rack, about 30-45 minutes. Once cooled, you can gently slice off the domed tops to create flat layers for stacking.

In a large bowl, beat the room temperature salted butter with an electric mixer on medium speed until light and fluffy, about 2-3 minutes. Gradually add the powdered sugar, espresso powder, and the cooled salted caramel until well combined.

Once the cakes are completely cooled, slice each layer in half horizontally, creating four layers total. Place one layer on a serving plate and spread a generous layer of espresso buttercream followed by a drizzle of the salted caramel. Repeat this process with the remaining layers. For the final layer, frost the top and sides of the cake with the remaining buttercream and let some caramel drip beautifully down the sides.