Step 1Line a baking sheet with parchment paper. Butter or grease the paper and set the sheet aside.
Step 2In a small, heavy pot, combine the sugar, water and corn syrup, stirring until the sugar has the consistency of wet sand. Place the saucepan over high heat and cook until the sugar dissolves and begins to boil. Do not stir the sugar, as this may cause it to seize.
Step 3While the sugar is cooking, combine the cream and butter in a saucepan over medium heat. Keep an eye on the sugar while you're heating the cream to keep it from scorching. Cook until the butter melts, stirring it into the cream. When the mixture has come to a simmer, remove from heat.
Step 4Continue to cook the sugar until it darkens to a rich caramel color, 7 to 10 minutes — the sugar will darken quickly and noticeably and will smell faintly nutty. (For lighter caramel, simply cook the sugar to a lighter color.) Swirl the pan as the sugar darkens to judge the true color of the caramel (the sugar may darken in patches if there are hot spots on the stove). Watch carefully, as the sugar can easily overcook at this point and burn.
Step 5As soon as the color is darkened, remove the pan from the heat and quickly add the cream mixture in a slow, steady stream. The sugar will bubble and steam as the cream is added; be careful as both the mixture and steam are very hot.
Step 6Bring the mixture to a simmer and cook just until a candy thermometer inserted reaches 248 degrees. Carefully stir in the vanilla. Remove from heat.
Step 7Pour the caramel into a heatproof and microwave-safe bowl and set aside until the caramel begins to cool and thicken. You want the caramel cool enough so the apples have a nice thick coating as they are dipped, but still warm enough that the excess caramel drips off somewhat easily. If the caramel cools too much, microwave the caramel to warm and thin as needed.
Step 8Dip the apples in the caramel, coating them on all sides and allowing any excess to drip back into the pot. Transfer the coated apples to the baking sheet, sprinkle with sea salt, and set aside to cool completely.