In a large, heavy saucepan, combine the butter, sugars, corn syrup, heavy cream, and salt. Bring to a roiling boil over medium heat. Turn to medium-low heat and boil for 5-8 minutes, or until thickened. Remove from the heat and stir in the vanilla.
Let the caramel cool to room temperature while you prepare the ice cream.
Ice Cream Base
Whisk together the milk, cream, brown sugar, corn syrup, salt, and vanilla. Place in the freezer for an hour until it is very cold. (Or chill it in the fridge for 3 -4 hours.)
Churn the ice cream in an ice cream maker according to the manufacturer’s instructions. It should be soft serve consistency.
In a freezer safe airtight container, quickly layer about a third of the ice cream, a fourth of the caramel, and a third of the salt. Repeat layers 2 more times. (You will have some caramel left over; save it for drizzling on top of the ice cream when you serve it.)
Place the ice cream in the freezer for 4-6 hours, or until it is firm enough to scoop. Serve with the caramel sauce on the side.
Notes
Flaky sea salt is the secret to this ice cream, so I highly recommend that you buy it. Regular table salt just doesn't work as well. In a pinch, you can use high quality sea salt.
A bit of corn syrup helps make softer, creamy ice cream and prevents ice crystals from forming. But if you prefer, you can leave it out.
The caramel can't be chilled, or it is too firm to spread on the ice cream. But because it is at room temperature, it does take longer for this ice cream to firm up in the freezer than most ice cream recipes.