Grease a 9-by-13-inch glass baking dish, and then line with parchment paper.
In a medium pot, bring the cream, corn syrup and vanilla to a boil. Turn the heat down and reserve on the back burner at a simmer.
Place the sugar in a separate large pot over high heat. Melt the sugar, stirring occasionally. Cook until the melted sugar reaches a light caramel color, swirling the pot to distribute the color, a few minutes.
Once golden in color, immediately stir in the butter and salt into the caramel, being careful, as it will bubble up.
Add the milk to the hot cream mixture, and then pour the mixture through a sieve into the caramel, being careful, as it will bubble up.
Cook the caramel until a candy thermometer inserted in the mixture reaches 260 degrees F. Then pour the caramel into the prepared dish and sprinkle with the salt. Refrigerate until set, 2 hours.