Place sugar and corn syrup in a 4-5 quart kettle and place over medium-high heat. Let the sugar begin to melt. The sugar will liquefy around the edge of the kettle. Begin to stir constantly to ensure the sugar melts evenly. Stir until the sugar is completely melted and dark amber in color. Remove from the heat.
Slowly stir in heavy cream in small amounts. The caramel will bubble vigorously, but continue adding cream until it is incorporated and the caramel is smooth.
1 cup heavy cream
Add the vanilla extract and rum, if using. Then add the cold butter and stir slowly to bring down the temperature of the caramel. Stir until completely smooth.