For the sauce
Add a small amount of the sugar to a medium hot pan set over medium heat and allow it to melt, then add the remaining sugar in small increments, allowing each addition to melt fully before adding the next. Continue this process until all sugar has been added to the pan. Cook to until a caramel develops. Remove from the heat and carefully, add the coffee and stir to combine. Put the espresso beans and cinnamon sticks in the caramel and simmer until the caramel is infused with the flavoring ingredients, about 15 minutes. Add the cream, stir in the butter and coffee liquor and mix until fully blended. The sauce is ready to be served, or it may covered and stored in the refrigerator for later use.