Pour milk into a medium-size bowl, and stir in tapioca.
Let stand while peeling and dicing apples.
Combine apples and water in the top of a double boiler.
Over direct heat, bring apple mixture to a boil.
Cook over medium heat, stirring often, for 5 minutes, or until apples are tender and water has evaporated.
Add egg yolk, honey, and cinnamon to milk mixture, and beat until well blended.
Pour into apples.
Set pan over simmering water, and cook for 15 minutes, or until thickened, stirring often.
Remove from heat, and stir in vanilla.
Let mixture cool slightly.
In a small bowl, beat egg white until stiff peaks form.
Fold egg white into warm pudding until well blended.
Chill pudding before serving.