Combine the sugar, water, and corn syrup in a 2-quart saucepan over low-medium heat. You can stir gently to help moisten the sugar crystals, but try to avoid getting sugar crystals up the sides of the pot.
Increase the heat to medium and continue cooking; the sugar should be at a low boil. DO NOT STIR WHILE THE SUGAR IS COOKING! You can swirl the pot gently to mix it and see the color of the sugar. Note the color of the sugar now so that you can watch for it to turn deeper golden when it's ready-- watching my video linked above will be helpful for this step.
Once the sugar turns a lovely medium-deep golden brown, IMMEDIATELY pour in the cream to stop the cooking. Turn off the heat. Now you may stir-- use a whisk to stir the sugar and cream together. It will look separated for 15 seconds and then come together.
When the caramel has come together, stir in the vanilla.
Add the butter and continually whisk until all the butter is incorporated, about 1 minute.
Stir in the salt and whisk until it is dissolved, about 1 minute.
Pour the warm caramel into a glass jar for storage; store in the refrigerator.