In a small bowl, whisk 1/2 cup of the milk with the cornstarch, vanilla and salt until smooth. In a large saucepan, combine the sugar with 1/4 cup plus 2 tablespoons of water and bring to a boil. Cook over moderately high heat, undisturbed, until a deep amber caramel forms, about 8 minutes. Remove from the heat. Very gradually whisk in the remaining 3 1/2 cups of milk, 1/4 cup at a time. Whisk over moderate heat until the caramel has dissolved. Simmer over moderately low heat until the mixture thickens slightly and deepens in color, about 10 minutes. Remove from the heat.