Combine 1/2 cup milk, half-and-half, and 1/3 cup sugar in a medium saucepan over medium heat. Cook, stirring constantly, for 2 minutes or just until mixture begins to simmer. Remove from heat, and add chocolate chips. Let stand for 5 minutes; stir until smooth.
Place chocolate mixture over low heat; return to a simmer, stirring frequently. Combine remaining 1/2 cup milk, cornstarch, egg, and egg white in a medium bowl; stir with a whisk. Gradually add 1 cup hot chocolate mixture to cornstarch mixture, stirring constantly with a whisk. Return the mixture to pan. Cook, stirring constantly, over low heat 4 minutes or until thickened. Stir in vanilla. Serve warm, or cover and chill. Garnish with chocolate shavings before serving.