First, prepare pudding.
In a large saucepan, stir flour, sugar, and milk together.
Stir continuously, cooking over medium heat, until bubbly.
Cook, while stirring, for two minutes more.
Remove from heat and stir about a cup of the milk mixture into the eggs gradually.
Add this egg mixture back into the saucepan with the rest of the milk mixture and cook until NEARLY bubbly, avoiding a boil.
Cook on low heat for two minutes more, stirring.
Stir in the vanilla, cover the surface of the pudding with plastic wrap, and chill until serving time.
Preheat the oven to 400 degrees F.
Combine the water, oil, sugar, and salt in a large saucepan and bring to a boil.
Add all of the flour and stir well with a wooden spoon until dough forms a distinct ball in the middle of the saucepan.
Cook a minute longer, while mashing and mixing the dough.
Remove from the heat and let rest to cool for two minutes.
Beat each egg into the dough, one by one.
Drop the dough, in smoothed mounds, onto a greased baking sheet and bake about 25 minutes.
For assembly, use a serrated knife to cut almost entirely around each puff three-quarters of the way up, to make a lid.
Scoop out any excess dough that may be left inside, and spoon some filling into each puff, replacing the lids after finishing.
Dust in a light coating of powdered sugar for presentation.
