Line a 9x9-inch pan with a parchment paper sling. It helps to spray the pan with a little cooking spray first to help the parchment paper stick in place.
In a large, heavy bottomed pot that is at least 4-quart capacity, combine the sugar, corn syrup, butter, salt, and 1 cup of the cream. Slowly heat over medium heat, stirring constantly with a wooden spoon or heat resistant spatula, until the butter is melted and the mixture comes to a boil.
Heat the remaining cream in a bowl in the microwave for 1-2 minutes until hot. Slowly pour the cream into the caramel mixture, stirring constantly until combined.
Continue to cook over medium heat, stirring frequently, until the caramel reaches 240-245°F on a candy thermometer. This can take quite a while! You can also test the caramel by dropping a spoonful in a bowl of cold water and shaping it into a ball. If it is pliable and soft but holds it's shape, then it is ready.
Remove the caramel from the heat and stir in the vanilla. Carefully pour the caramel into the prepared pan (it's very hot so don't get any on your fingers!). Do not scrape the bottom of the pan.
Let the caramel cool completely for at least 8 hours or overnight before cutting into small squares.
When the caramel is set, lift the parchment paper sling out of the pan and use a sharp knife or pastry cutter to cut out the caramels. Wrap individual caramels in 5-inch square pieces of wax paper, twisting the ends to secure.