Lightly butter a 9x13-inch pan and line it with parchment paper, letting the excess come up the sides of the pan. Lightly butter the paper.

In a large heavy-bottom saucepan (make sure it’s a 4 to 5-quart capacity), combine the sugars, corn syrup, butter, and salt. Place over medium heat and stir constantly until the butter and sugars have melted and the mixture starts to bubble but it isn't boiling hard, about 5 minutes. (If any sugar crystals remain on the sides of the pot, use a wet pastry brush on the inside of the pot to brush off and dissolve them.)

While whisking, slowly pour in the cream (this should take about 1 minute). Attach a candy thermometer and bring to a boil, stirring constantly until the mixture reaches 245°F to 248°F, 20 to 25 minutes. (You can also drop a small ball of caramel into a glass of cold water. If it forms a firm ball but is still moldable, it’s done.)

Immediately remove from the heat and carefully stir in the vanilla. Pour into the prepared pan. Let the caramels cool completely, about 4 hours. If desired, sprinkle flaked sea salt on top of the caramels after cooling for about 30 minutes. (To speed up the cooling, you can refrigerate the caramels until hardened. Just remove the slab about 30 minutes before slicing as the cold caramel can shatter or break when cut.)

Remove the cooled slab from the pan and discard the parchment paper. Cut into squares and wrap in wax paper. Caramels will keep at room temperature for up to 1 month or refrigerated for several weeks. They should be enjoyed at room temperature.