Using a 9×13 baking dish, line it with foil or parchment paper. Take some butter and generously grease the foil/paper. Set the dish aside.
Place the salted butter, light brown sugar, granulated sugar, and light corn syrup into a dutch oven or medium to large pot. Place over medium heat and constantly stir until the sugars melt and boil for about 10 minutes.
Once the mixture is at a boil, very, very slowly add the heavy cream. I mean slow, add about 1 teaspoon at a time, and constantly stir; you want the mixture to be boiling the entire time. This step can take 20-30 minutes to add all the cream.
Continue to boil the mixture and stir constantly. Scrape the bottom and sides of the pot while doing so until the mixture comes to 240-245°F. I like to use a candy thermometer for this.
Take off the heat and stir in the vanilla.
Immediately pour the caramel into the prepared baking dish. Let it sit out at room temperature until completely cooled.
Refrigerate until hardened, 4-6 hours or overnight is best. Take the caramels out by lifting the foil or parchment paper out of the dish and then cut them into small pieces.
Sprinkle generously with sea salt and serve immediately.