In a medium-sized saucepan, pour the water over the sugar and cook this mixture over medium heat, until the color is a dark amber. This should take about 15 minutes. If you want a slightly more mellow -- less intense -- caramel, go for a lighter amber color, with should take less than 15 minutes. Do not ever stir this mixture while it's cooking!
Once you've achieved the desired color, reduce the heat to low and carefully whisk in the butter, heavy cream, and sour cream. Stand back! When you add these ingredients, the mixture will likely hiss and bubble up. Continue to whisk until it's smooth. Then mix in the espresso powder, stir to blend, and remove from the heat. Let is sit at room temperature until you're ready to use it.
Preheat oven to 375°F.
Add the sugar and mix well. Then add the eggs one at a time, whisking well after each addition. Pour the batter into the prepared pan and bake in the preheated 375°F oven until cake has risen and the top and edges have formed a thin crust, about 25 minutes.
Please note that the prep time does not include letting the caramel set after you've added it to the brownies, should you choose to do so.
This caramel portion of the recipe can be made up to at least a week ahead of time. Keep it stored in the refrigerator and warm it to soften.