Line the bottom and sides of an 8-inch square pan with parchment paper. Brush the paper lightly with oil
Combine the heavy cream, butter, and salt in a small saucepan and heat over low heat until the butter is melted. (This can also be done in the microwave) Set aside.
Combine the sugar, water, and corn syrup in a heavy 4-quart saucepan and put over medium-high heat. Stir occasionally until the sugar has dissolved. Cover and allow to cook, untouched, for 5 minutes.
Remove the lid and check the temperature. When the sugar mixture reaches 230 °F, reduce the heat to medium and continue to cook, without stirring, until the syrup has reached 300 °F.
At this point, there is less likelihood of crystallization, so gently swirl the pan to eliminate hot pockets. Continue cooking and swirling the pan as the syrup increases in temperature and changes color. When the syrup reaches a deep amber or warm golden brown remove the pan from the heat. (This will be shortly before it starts to smoke and smell burned.) Let cool for 2 minutes.
Carefully incorporate the cream and butter mixture into the pan. Be careful, it will bubble up violently. Add the vanilla and stir to combine. Return the caramel to medium (to medium-low) heat and continue cooking, stirring frequently, until it reaches 250 °F. (Just past Firm Ball on a candy thermometer)
Remove from the heat and pour into the parchment-lined pan, tap gently to release air bubbles.
Cool on a cooling rack for 30 minutes, and then sprinkle evenly with the salt. Continue cooling on the rack for an additional 3 1/2 hours. Cut into 1-inch pieces and wrap individually in parchment. Store in an airtight container for up to a week.