Add tapioca and water to a medium saucepan. Let soak for 30 minutes.
In a small bowl beat together egg yolks and pumpkin.
Add mixture to tapioca along with milk, 1/4 cup sugar, salt, cinnamon, and nutmeg.
Simmer tapioca mixture over low heat for 10-15 minutes, uncovered, until it starts to thicken.
Meanwhile, beat egg whites with 1/4 cup sugar to soft peaks (a cold clean bowl helps). Whip 1 cup of the hot tapioca mixture into the egg whites. Then fold the egg white mixture back into the tapioca pot. Stir over low heat for another few minutes. Remove from heat and let the tapioca cool for a few minutes. Then stir in vanilla. Serve tapioca warm or chill for later. It’ll set up more if you let it chill. Serve with whipped cream.