In a medium bowl, combine the graham cracker crumbs and melted butter, mixing until well blended. Press this mixture firmly into the bottom of a 9-inch springform pan to create the crust. Bake for 10 minutes, then remove from the oven and set aside to cool.
In a large mixing bowl, beat the softened cream cheese and ½ cup of sugar using an electric mixer until smooth and creamy.
Add the eggs one at a time, mixing thoroughly between each addition to ensure a smooth batter.
Mix in the vanilla extract, cooled brewed coffee, and espresso powder until fully incorporated.
Gently fold the sour cream and ½ cup of caramel sauce into the batter.
Pour the cream cheese mixture over the cooled crust in the springform pan.
Bake in the preheated oven for about 60 minutes, or until the center is set but still slightly jiggly.
Turn off the oven and leave the cheesecake inside for 1 hour to cool gradually, which helps prevent cracking.
Remove the cheesecake from the oven and let it cool completely at room temperature before refrigerating for at least 4 hours (or overnight) to set fully.
Before serving, drizzle additional caramel sauce over the top and garnish with whipped cream if desired.