Lightly beat egg with a fork in a medium bowl.
Combine 11/2 cups milk, 1/3 cup sugar and salt in a medium saucepan; bring to a simmer over medium heat, stirring occasionally.
Meanwhile, whisk the remaining 1/3 cup sugar, cocoa and cornstarch in a medium bowl. Whisk in the remaining 3/4 cup milk until blended. Whisk the simmering milk mixture into the cocoa mixture. Pour the mixture back into the pan and bring to a simmer over medium heat, whisking constantly, until thickened and glossy, about 3 minutes. Remove from heat.
Whisk about 1 cup of the hot cocoa mixture into the beaten egg. Return the egg mixture to the pan and cook over medium-low heat, whisking constantly, until steaming and thickened, about 2 minutes. (Do not boil.) Whisk in vanilla. Serve warm.