Mix the sugar and cornstarch together in a sauce pan. Gradually add the milk and stir constantly. Bring to the boil and continue boiling the pudding for 2 to 3 minutes, stirring constantly. Remove from the heat.
Beat the eggs in a little cold milk (about a tablespoon) and add to the sugar mixture stirring all the while. Return the pan to the stove and cook over gentle heat till the mixture thickens slightly. Do not boil. Add vanilla to taste.
Pour into a heat proof serving dish. Generously sprinkle the pudding with granulated sugar and brown under the broiler.