Add sugar, water, and corn syrup to a medium saucepan, and bring to a boil over high heat, stirring until the sugar is dissolved. As soon as the mixture comes to a boil, clamp on a lid, and cook for 6 minutes. Remove the lid and check the color and texture of your caramel. It should still be clear, and the bubbles should be large. Continue to cook with the lid off. As the mixture starts to brown and darken in color, gently swirl the pan to mix. Do not stir here. Once the mixture is a dark amber color (not black), take the saucepan off the heat, and carefully add the heavy cream. The mixture will violently bubble up. Let the caramel settle for 1 minute, then stir in the bourbon and flaky sea salt. If the caramel has seized up a bit (hardened), simply return the saucepan to a low flame, and stir just until the mixture smooths out. Let mixture cool completely before adding to the milkshake. Store in the refrigerator for up to 1 week.