Combine the sugar and water in a heavy saucepan over medium-high heat. Bring to a boil, stirring occasionally, until the sugar is dissolved. Once the sugar is dissolved and the syrup is boiling, cook without stirring for 5-10 minutes or until the syrup is a rich amber color. Remove the pan from the heat and carefully and slowly pour in the coffee. Let stand for 2 minutes. Return the pan to the stove over low heat. Whisk the mixture until it is smooth. Remove from heat. Add the butter to the pan and stir. Add the milk while whisking constantly. Stir in the vanilla. Let the sauce cool completely. Store in the refrigerator in a covered container for up to 7-10 days.