For pudding: Put the cold water into a large mixing bowl and sprinkle the gelatin over it. Allow it to set for 1 minute. Add the hot water, lemon juice, and sugar and stir until the gelatin and sugar dissolve. Refrigerate the pudding until it is thick, but not firm (6 to 8 hours). In a separate bowl and with an electric mixer, beat the egg white until it forms soft peaks. Remove the gelatin from the refrigerator and fold the egg white into it. Return to the refrigerator until ready to serve.
For sauce: In a saucepan, stir together the egg yolk, cornstarch, and sugar. Add the milk and cook over medium heat, stirring constantly until slightly thick. Remove from the heat, stir in the vanilla, and set aside to cool. Spoon the sauce over individual servings of pudding.