Make the Caramel Custard
In a medium saucepan, combine the sugar with 1/4 cup of water. Cook over moderate heat, swirling the pan and brushing down the side with a wet pastry brush, until the sugar dissolves. Cook undisturbed until a deep-amber caramel forms, about 5 minutes. Remove the saucepan from the heat and whisk in the butter; be careful, it will boil vigorously. While whisking constantly, slowly drizzle in the warm milk until blended; the mixture may separate a little. Slowly whisk in the egg mixture and bring to a boil, then cook over moderate heat, stirring, until thickened, about 2 minutes. Stir in the vanilla and sea salt. Strain the caramel custard through a fine sieve into a heatproof bowl. Press a piece of plastic wrap directly onto the surface of the custard and refrigerate until completely chilled.