Grease a 9×13-inch pan with a light layer of butter, then line it with parchment paper, allowing the extra paper to hang over the sides. Lightly butter the parchment as well.
In a large heavy-bottom saucepan, about 4 to 5 quarts in size, combine the sugars, corn syrup, butter, and salt. Set the pan over medium heat and stir constantly until the butter melts and the sugars dissolve. Let the mixture heat until it begins to bubble gently but is not at a full boil, about 5 minutes. If you notice sugar crystals clinging to the sides of the pan, use a damp pastry brush to wipe them down and help them dissolve.
While whisking steadily, slowly pour in the cream, taking about 1 minute to add it all. Clip on a candy thermometer and bring the mixture to a boil, stirring nonstop until it reaches 245°F to 248°F, about 20 to 25 minutes. You can also test it by dropping a small bit of caramel into a glass of cold water. If it forms a firm ball that you can still shape, it is ready.
Take the pan off the heat right away and carefully stir in the vanilla. Pour the caramel into the prepared pan and let it cool completely, about 4 hours. If you like, sprinkle flaked sea salt over the top after it has cooled for around 30 minutes. To speed things up, you can place the pan in the refrigerator until firm. Just take the slab out about 30 minutes before cutting, since very cold caramel can crack or break when sliced.
Lift the cooled caramel slab out of the pan and peel away the parchment paper. Slice into squares and wrap each piece in wax paper. Store the caramels at room temperature for up to 1 month, or keep them in the refrigerator for several weeks. For the best texture and flavor, let them come to room temperature before enjoying.
