Place a piece of parchment paper in an 8x8-inch or 9x9-inch square pan, then lightly grease the parchment paper. Set aside.
Add the butter, granulated sugar, brown sugar, heavy cream, water, and corn syrup to a large heavy-bottomed pot.
Heat over medium heat and stir constantly with a rubber spatula or wooden spoon until everything is melted together.
Bring the mixture to a boil, then reduce the heat to medium-low and add a candy thermometer to the pot.
Let the mixture simmer, stirring every 10 minutes, until the temperature on the candy thermometer is between 245-250 F (it will take about 45 minutes).
Turn off the heat and stir in the fine sea salt and vanilla extract.
Carefully pour the hot caramel into the prepared baking dish.
Allow the caramel to cool for about 20 minutes, then sprinkle half of the flaked salt over the caramel.
Let the caramel set at room temperature for at least 2 hours, then sprinkle the remaining salt over the caramel and carefully peel back the parchment paper on the sides of the caramel block.
Use a sharp knife to cut the caramel into squares.
Serve caramel candies right away. Wrap leftover caramels individually in wax paper or candy wrappers, and store in an airtight container.