Line an 8 × 8-inch baking pan with parchment paper and lightly grease it with butter. Set aside.
Add sugar, heavy cream, corn syrup, and butter to a heavy-bottomed saucepan. Heat over medium heat, stirring gently until the butter melts and the sugar dissolves.
Bring the mixture to a gentle boil. Attach a candy thermometer to the pan and continue cooking, stirring occasionally, until the caramel reaches 245°F–250°F (118°C–121°C).
Remove the caramel from the heat immediately once it reaches the correct temperature. Stir in vanilla extract and sea salt carefully, as the mixture will bubble.
Pour the hot caramel into the prepared pan and smooth the surface with a spatula. Let it cool completely at room temperature for about 4 hours, or until firm enough to cut.
Lift the caramel from the pan using the parchment paper. Cut into small squares using a sharp knife lightly coated with butter or oil.
Wrap each caramel piece individually in parchment paper or wax paper and store in an airtight container.